One of the things that were very unnatural for me while learning English was the "th" sound. Do all native English speakers actually pronounce it this way or does it vary between accents (Canadian, US, Australian, UK islands)? Does it actually stand out if I pronunce it as "f"?
Most native English speakers you hear will effortlessly pronounce the th digraph you're having trouble with. While there are some dialects of English that pronounce it /d/ or /t/ or /f/ depending on position, standard pronunciations in the US and UK pronounce it "normally" and that is what you should strive to emulate if you want to sound like a native speaker.
There are phonemes in every language that non-native speakers have trouble with, and English is no exception. This is the advantage of growing up speaking a language from childhood. And by this I mean from very early childhood, what most people would consider the pre-verbal period of a baby's life.
9-month-old babies are aware of the phonemes in their own language as they start to use both prosodic and phonotactic cues to discriminate individual speech sounds of their language
Some studies have shown that unless a baby hears its language's phonemes in its first six months of life, it may never code for them at all.
The point is, yes, it's hard to duplicate certain sounds in another language. You may never pronounce those sounds perfectly. But unless you make the effort, your pronunciation will always mark you as foreign* and, worse, you may have trouble communicating with native speakers.
In response to a comment I'm including further information about the critical nature of language exposure in the early months of life:
At birth, infants are prepared to learn any language. For example, an American baby adopted by an Inuit-speaking Eskimo family would grow up speaking fluent Inuktitut and have no trouble saying words such as qikturiaqtauniq ("mosquito bite"). However, even before their first birthdays, babies begin to lose the ability to hear the distinctions among phonemes in languages other than their own. By around the age of six months, babies have already begun to hear the sounds of their own language in the same way that adult speakers do, as Patricia Kuhl and her associates (1992) have shown in their research.
It's worth noting that they say babies before their first birthdays are beginning to lose "the ability to hear the distinctions among phoneme in languages other than their own." Not that they've lost it, but that the longer a child goes without hearing those distinctions and, consequently, producing them itself, the harder it will be for that child to reproduce all the language's sounds. By the time one reaches adulthood, it can be a monumental task.
Anecdotally, my own name, which is Germanic and contains the ü sound in German, is extremely difficult for me to pronounce fluently; and a word like Brüder, with its combination of the glottal /r/ immediately followed by the ü, is well-nigh impossible for me—even though I worked in Germany for a time and acquired a fair bit of fluency. It was always a source of chagrin for me, especially when I would hear my coworkers pronounce my name flawlessly and without effort.
* And in case you think that it's all right to use those non-standard sounds produced by dialectical speakers, be aware that even to sound like them you would have to master the whole range of their pronunciations as well, and be able to use those when appropriate, which would be just as big a task (if not bigger) as learning the standard version.
All "standard" accents maintain the sounds /θ/ and /ð/
It will definitely stand out if you can't pronounce the "th" sounds (there are two, the voiceless version /θ/ and the voiced version /ð/). For adult native English speakers with a standard accent, it comes completely naturally and it doesn't take any special effort to pronounce these sounds. A non-native speaker aiming to speak without a strong accent definitely needs to learn how to pronounce these sounds. However, it won't generally destroy your comprehensibility if you replace them with other sounds.
There are some native accents where these sounds are replaced by others, and English speakers are also familiar with non-native speakers replacing these sounds.
The most common replacements for /θ/
The most common native English replacements are turning the fricatives into stops (/θ/ to /t/ , and /ð/ to /d/) or fronting them to a labiodental position (/θ/ to /f/, and /ð/ to /v/).
For /θ/, fronting to /f/ is probably the most common replacement among native speakers, although it is still not that common. It occurs in Cockney and certain other (mainly viewed as lower-class) British accents. According to Wikipedia, if I'm reading it correctly, in African American Vernacular English it usually occurs only in non-word-initial positions. So an example of this would be "bath" pronounced as "baf", or for a Cockney speaker, "thing" pronounced as "fing". I believe this substitution also occurs fairly frequently in children's speech.
Stopping of /θ/ to /t/ seems to occur somewhat sporadically in native English accents, but it is still a fairly familiar sound change and it wouldn't be surprising to hear e.g. "ting" for "thing" from a non-native speaker. It does contribute a lot to the perception of an accent as foreign, though, and it might cause some confusion with certain homophones (e.g. "bath" and "bat", in American English).
Replacement of /θ/ with /s/ is also known, although as far as I know it occurs exclusively in foreign accents.
The most common replacements for /ð/
For word-initial /ð/, the most common pronunciation in accents without a separate /ð/ phoneme is probably with a stop /d/. I.e. "that" is pronounced like "dat", "the" is pronounced as "da", "this" is pronounced as "dis". The pronunciations with /d/ are generally stigmatized, so it will stand out if you use them and it might sound silly, like you are putting on an accent that isn't associated with your linguistic background. In America, stopping word-initial /ð/ to /d/ is associated in particular with African American Vernacular English.
However, note that even in accents with a distinct phoneme /ð/ that can occur word-initially, this sound is frequently subject to some degree of phonetic assimilation: when it comes directly after another consonant, it tends to be pronounced differently depending on what the preceding consonant is. After a nasal consonant like "n", it may become nasal (e.g. in sequences such as "in the"); after a stop or affricate, it may become more like a stop or affricate (for more information like this, see "The stop-like modification of /ð/: A case study in the analysis and handling of speech variation", by Sherry Y. Zhao, 2007).You probably never have to aim to produce assimilated forms, but you should be able to recognize them from native speakers.
Another possible replacement for /ð/ is the sound /v/ in some accents (e.g. "brother" = "bruvver", "another" = "anuvver"). This is the same kind of "fronting" as the replacement of /θ/ with /f/, and these phenomena seem to often go together. Fronting of /ð/ to /v/ is associated with Cockney and certain other (mainly viewed as lower-class) British accents. I had the impression that it occured word-medially but not word-initially, but a comment by AkashM indicates that it can occur even at the start of a word:
plenty of the speakers that have it medially also have it initially (source: personal experience). "there's no way" -> "vere's no way", "the main thing is" -> "ve main fing is". – AakashM
It is generally understandable to speakers of other accents, but it definitely stands out and is associated with particular regional varieties of speech.
Replacing /ð/ with /z/ will probably be understood, but it definitely sounds foreign. (E.g. pronouncing "this" as "zis", "that" as "zat", "other" as "uzzer".) I don't know of any native English accent that has this. Using /z/ in place of /ð/ is a standard feature of a stereotypical strong French accent.
There is African-American Vernacular English. The th sound appears to be used more rarely (if ever):
When occurring in the beginning of a word, the th- sound is pronounced as a d- sound.
example: this, they, that --> dis, dey, dat
Within a word, -th (unvoiced) is frequently pronounced as an f sound. This also occurs at the end of the word in certain environments.
example: nothing, author --> nuffin, ahfuh
The voiced -th may be voiced as a v sound.
example: brother --> bruvah
Of course, speakers of AAVE can also speak with a General American accent, which would include the th sound.
As hatchet points out, Cajun English is another example:
Voiceless and voiced /th/ replacements occur frequently in the speech of non-standard speakers, and the Cajuns are no exception. In fact, the replacement of the /th/ sounds with a /t/ or a /d/ sound is another source of the numerous jokes and imitations of Cajun speech made by others (and sometimes by Cajuns themselves, as in the “Cajun Night Before Christmas” recording made by Jules D’Hemecourt). Although many southern English and African American English speakers use an /f/ or a /v/ in place of the /th/ phonemes, both Creole and Cajun English speakers use the voiceless and voiced alveolar stops /t/ and /d/. Many bilingual French-Canadians exhibit this same linguistic behavior with regard to the /th/ phonemes, while standard French speakers tend to use an /s/ or a /z/ in pace of a “th” sound.
This quote also shows you that southern English is similar to AAVE, while Creole and French-Canadian English are similar to Cajun English with how they pronounce th.
The standard accent for Irish native speakers of English does not use /θ/ and /ð/. In novels Irish people are often depicted as invariably replacing /θ/ and /ð/ by /t/ and /d/, but that is described as a misconception in this account of the phonology of Irish English produced by the University of Duisburg-Essen. Under the heading "Misconceptions about Irish English", it says,
1) The Irish pronounce the th in thinker like the t in tinker.
This is generally untrue. In non-vernacular speech in the south of Ireland a strict distinction is maintained between a dental [ṯ] (as in Swedish tala ‘speak’ or Italian notte ‘night’) and an alveolar [t] (as in English tall or not) so that the words thin and tin are not homophones. The Irish are very sensitive to the shift from dental to alveolar stop and they regard the use of the latter in the THIN lexical set as a sign of strongly vernacular speech.
In northern Ireland the ambi-dental fricatives of more standard English are found so that thanks is [θæŋks]. This fricative is sometimes found as a spelling pronunciation with southern speakers in word-final position.
I think it is fair to say that middle class standard Irish English replaces /θ/ and /ð/ with voiceless and voiced dental stops, /t̪/ and /d̪/, but most sorts of working class Irish English, in addition to some regional dialects irrespective of social class, replace them with the equivalent pair of alveolar stops, /t/ and /d/.
This broadly agrees with the Wikipedia account of the consonants of Hiberno-English.
I have heard Newfoundlanders in Canada replace the "th" sound with "t" as "tree" instead of "three" or "Tursday" instead of "Thursday."
But to help you learn to pronounce it properly, I am ESL teacher and I often used the attached image to help my students learn the correct placement of the teeth and tongue. Look at the image while saying "th" words. Use a mirror to practice correct placement. It may seem silly but this really helped my students!
As the other answers state, yes, the majority of native English speakers do/can pronounce the "th" sound (be it as in "the" or "three").
Though this isn't the case from the word go. Children who are learning to talk will usually pronounce "three" as "free" and pronounce "the" with sounds they find easier to make (until they are able to make "th") such as "da", "va" and "za".
As another answerer mentioned, African-American vernacular is often distinctive by its replacement of "th" with "d-". However, that is not due to an inability to make the "th" sound.