I open 1D, partner bids 1S, I bid 2NT, opponents have remained silent - partner bids 3D? Is that forcing?
This is very system dependent. See this article for example for a long discussion of various options.
Standard, everything other than pass or a game jump is forcing. Remember you have 19 and partner has 5-6 at least usually, so you’re probably in a game. I would at least play Gerber here (so 4C is not natural) but 3 level bids are forcing and usually natural, other than 3NT of course which is to play.
The most common conventions I see here are to play various puppet options - basically every bid by responder transfers to the next higher suit at the 3 level (3c-3D-3h transfer to 3D-3h-3s), and either that is the weakest option (responder passes the transfer) or the stronger (if responder bids on). Jump to game is middle. But it really depends on your tolerance for various systems - it can get very complex, as is appropriate for a circumstance where you could have anywhere from a part score to slam, and one partner knows which.
The question is why you’d want 3D to be nonforcing. Responder is totally 100% in charge in the auction at that point, so it’s not a question of forcing or not - it’s responder either tells you to bid again or tells you to pass. You could make 3 level bids sign off, but that takes up a lot of space; you’d rather not if you can avoid it. That’s why you use puppet bids there - it lets responder double dip and have both signoff and exploration bids both available for the same bidding space.
Agreeing with Joe here; standard is "everything is game forcing except pass." He gives the reason why non-game forcing calls are of limited value. Specifically in your case, the hands that are "5-6 points, 5 spades and 4 diamonds (or 4=4, or 4=5), 3D is clearly better than 2NT" are very few. Also, it's very rare that the weak hand knows that NT isn't good.
The other half of the story, why you would want "everything is game forcing", is that decent 10 counts (or better) with a good suit or suits are interested in slam, and want the space below 3NT (and below 4S) to work out if it's safe to go past. How do you get to 6D if 3D is passable?
The other hand of interest is the "checkback stayman" hand that bids "the unbid minor" - 5 spades and 4 hearts. Joe's puppet scheme would be a useful equivalent, but still has to be game-forcing, as you would be bidding 1D-1S; 2NT-3D(hearts); 3H-3S(5=4) and if partner has 4 hearts, she has to bid game. Hands with 5 spades without 4 hearts also go through checkback, and pretty much have to be game-forcing in case partner is 2=4=4=3 or the like.
So, it's a matter of what hands you want to cater for; and "wisdom" is that the hands that want to stop in 3 of a suit (rather than 2NT) are very small, and the cost of not catering to them is also small; the hands that need to investigate (slam, or two suits) are larger, and the cost of not being able to cater to them (or requiring "all slams go through Gerber") is high.
I've played a system in the past that had a good 16-bad 19 2NT rebid; we "sacrificed" 3C as a "minimum range" signoff signal. There were lots of rules about how to accomplish the signoff, and 3C-then-3NT was the "strong 3C bid", and it worked. However, it was rare even with the "could be a good 16" worry, and whether the complication was worth the little edge we got is arguable. Over a standard 18-19 rebid, it's even more arguable.