Actually, it's not the /th/ that usually gets dropped. It's the /i/. First, in contractions:

What's there?
It's there.

And secondly, at the beginning of a sentence the /i/ is often dropped, so that the sentence actually starts on a /z/ sound:

zthere any coffee left? [Is there any coffee left?]

The /z/ sound is very brief and after experimenting with this for a while, it sometimes sounds like /sz/ (szthere). You start with the tongue almost stopping the breath at the front of the palate (the place where you would start a /t/ or /th/ sound, let some air through which begins a sibilance, add voicing which turns it into a /z/ for a few milliseconds, then slide the tongue up behind the teeth very rapidly to form the vocalized /th/.

In cases where the first syllable is not dropped, you do the process mentioned above except starting with a full vocalized vowel /i/, then moving the tongue to create the /z/ and so on. It's not hard for native speakers.

I know that the words there, their, and they're are homophones

Yes.

but I can't help but think that their has a slight /j/ between /e/ and /r/ (/ðer/).

It’s possible, but no more so than for there and they’re. I don’t think you’re imagining things, but I also wouldn’t recommend taking any special efforts to pronounce this word this way.

For American English speakers, there is generally no phonemic contrast between the vowel in there/their/they’re and the vowel in they/say/fade/male/mail. (This holds regardless of whether we’re talking about an accent with or without the Mary-marry-merry merger.)

But phonetically, you’re less likely to hear a /j/-like glide when the vowel is followed by /r/ with no intervening word boundary, and more likely to hear a glide when the vowel is word-final, followed by a vowel, or followed by most other consonants, like /p b m t d n k g s z f v θ ð tʃ dʒ/. For me, the /j/-like off glide of the vowel /e/ is also suppressed to some extent before a following /l/ (which I pronounce as “dark l”, [ɫ]), but I don’t know how common this is or the details of how it works (I do distinguish pairs like fail and fell).