Haley found the remark disrespectful and felt she couldn't keep quiet, the sources said.
I heard someone read this, and I can hardly hear "the" between "found" and "remark".
*"Haley found remark disrespectful" is ungrammatical, so you'd be unlikely to hear a native speaker say it.
What you're likely hearing is the "the" being reduced, likely with almost no vowel sound. Sometimes the "th" might sound more like a "d", as well.
e.g., a native speaker might pronounce it like either of the following, in casual speech:
Haley found th' remark disrespectful
Haley found d' remark disrespectful
In such cases, I'd often expect the first "e" in "remark" and/or "disrespectful" to be heavily reduced, as well.
In extreme cases, you might even get the "d" in "found" dropped, so under the right circumstance, you could have a pronunciation which sounds like:
Haley foun' d'remark disr'spectful
Where "d'remark" would be "the remark" in well-enunciated speech, and "foun'" would be "found".
In my dialect (British, West Midlands), it actually comes out as:
"Hayley foun' the remark disrespectful."
The D in found is very, very heavily reduced in this sentence. In fact, forcing myself to enunciate the D as well as the "The" seems really unnatural. Perhaps this is why a non-native speaker could miss the different sounds?
But regardless, the grammatically correct sentence definitely includes the "the" and I can't imagine somebody omitting the word fully even in casual speech.
Native speakers don't "omit" the entire word, but many of them will strongly de-emphasize the vowel sound in the and essentially merge it with the start of the next word. It's not quite an elision of the syllable, because you can still hear it, but it's very fast and nearly omitted.
Typically this happens if the following word starts with an unstressed syllable. For example, a native speaker would likely sound like they were saying:
I found the entire exchange disrespectful.
I found the statement disrespectful.
I found th'remark disrespectful.
Others have already given good answers, but for what it is worth, I'm an U.S. speaker and would naturally say
"Haley foun' the remark disrespectful"
just like @Psiloc mentioned. But I would guess many Americans would also say
"Haley foun da remark disrespectful"
where "the" almost sounds like "da" attached to "found".
To add to the dialect versions, northern English dialects will often pronounce "the remark" as "ut-remark" with a glottal stop (spelled "t'remark" if you want to write it down). This is very easy to mishear.
It's standard for Yorkshire, parts of Lancashire, and parts of Derbyshire. You won't hear it further south, and when you get further north (up into Cumbria, Northumberland and Tyneside) then you have different accents again which also don't do this.
We definitely say it. As other answers have pointed out, the 'd' and the 'th' can get blended together, and the 'e' doesn't have much emphasis on it. But you can always hear the syllable, no matter how mangled the pronunciation is. Put simply, no matter how fast we're speaking, it takes us longer to say "found the remark" than "found remark".
It's been obliquely mentioned a few times here, but I thought it might be worth spelling out:
Not only is the reduced to th' when a native-speaker says this, but found is also reduced to foun'. A native-speaker will typically pronounce the D on "found" only if the following word begins with a vowel.
So when you're listening, you're merging foun' with th' to make something like founth (which sounds very much like found), and therefore it sounds to you like there is no the.
A native speaker would pronounce "found remark" as "foun' remark", and there would be no d/th sound between the words.
Subtle differences in the sounds formed are not necessarily conveyed to the listener. Try this test:
Choose a quiet place and native speaker. Speak the word facts then the word fax.
I tried this and my mouth formed the two words differently, but the listener could not hear the difference.
So although I would pronounce the phrase foun' d-th' remark, the listener will not hear the subtle changes the mouth makes switching from d to th.