In answering this question it is helpful to have a basic understanding of the way different sounds are made. There are two opposing forces acting here: the reduction of energy expenditure, and the necessity of comprehensibility. Certain sound combinations require more energy or more effort to pronounce correctly in terms of moving the tongue around the mouth, tensing and relaxing the vocal chords, and many other factors, leading to an attempt to reduce the energy required, which ends up changing the actual sounds being made. This reduction is limited by comprehensibility. If the sounds change so much that what is being said is no longer understood, more effort is required to make the sounds clearer.
I've taken the following chart of English consonants from Wikipedia:
Labial Dental Alveolar Post-alveolar Palatal Velar Glottal
Nasal m n ŋ
Plosive/ fortis p t tʃ k
affricate lenis b d dʒ ɡ
Fricative fortis f θ s ʃ x h
lenis v ð z ʒ h
Approximant l r j w
Note that the final sound of "or" is /r/ (or more accurately /ɹ/). American accents tend to be rhotic, although some are non-rhotic. Very simply, rhotic accents pronounce the "r" at the end of syllables, and non-rhotic accents don't. Non-rhotic accents are more commonly found in the UK, and the accents of Australia and New Zealand are entirely non-rhotic.
Note also that /r/ is in the row labelled "Approximant". The approximants /l/ ("let"), /r/ ("red"), /j/ ("yet"), /w/ ("wet"), are all treated as consonants at the start of a syllable, but can sound like a vowel, or "colour" the vowel sound at the end of a syllable:
- toll (US: /toʊɫ/, /tɔl/; UK: /təʊl/, /tɒl/; Canada: /toʊl/, /tɑl/ - /ɫ/ is "dark l", a "w"-like "l" found at the end of syllables for some speakers. Note how many of the pronunciations include /ʊ/, a "w"-like vowel, see "tow" below)
- tore (rhotic: /tɔɹ/, /to(ː)ɹ/; non-rhotic: /tɔː/, /toə/)
- toy (/tɔɪ/ - /ɪ/ not /j/)
- tow (/toʊ/, /təʊ/ - /ʊ/ not /w/)
For the word "tore", in the non-rhotic accents either the vowel is lengthened, or it is changed from a monophthong (single vowel sound) to a diphthong (two different consecutive vowel sounds).
The /h/ sound is easier to say immediately after a vowel than a consonant because the mouth, tongue, etc. are already closer to the correct positions to pronounce /h/, requiring less effort. In careful speech for someone who doesn't use h-dropping, the /h/ will always be pronounced, e.g. in "be his friend" and "want her money". As speech becomes more rapid and less careful, the /h/ will start to be dropped after certain sounds roughly in this order:
- the plosives (/p/, /b/, /t/, /d/, /tʃ/ ("ch" as in "chat"), /dʒ/ ("j" as in "jet"), /k/, /ɡ/)
- the fricatives (/f/, /v/, /θ/ ("th" as in "thing"), /ð/ ("th" as in "this"), /s/, /z/, /ʃ/ ("sh" as in "shed"), /ʒ/ (as in "measure"))
- the nasals (/m/, /n/, /ŋ/ ("ng" as in "sing"))
- the approximants (/l/, /ɹ/ ("r" as in "red", /j/ ("y" as in "yet"), /w/)
- finally the vowels
This analysis is approximate only, actual speech may vary.
That is, in the fastest, least careful speech, the /h/ will be dropped after an "r" (rhotic) or an r-coloured vowel (non-rhotic). The /h/ can also take on various strengths, being strongly, weakly, or not pronounced as speech goes from most careful to least careful. This is not something that happens consciously, and should not be deliberately imitated in regular speech by a non-native speaker. As a non-native speaker, your mouth has been trained to make the correct sounds in your native language, which are different from the sounds in English. You need to put in more effort to overcome this difference anyway, so trying to reduce your effort by imitating "h"-dropping is not a good idea. As your English improves with practice, you will pick up these nuances automatically and subconsciously.