In a training video of a full ring multi table tournament, the trainer talks about having to call if a certain player pushes all-in. It was in the context of deciding whether to 3-bet and the blinds contained players with stacks of around 17-20bbs.

At what stack size of opponent who pushes pre-flop is it a mistake to fold to after three betting?

(Apologies if I have worded this rather clumsily).

This depends on a lot of stuff. The more chips you have and the further from the bubble you are, the more you can just look at the odds you're getting and make the most +ev decision based on that.

I think it's generally a mistake to 3 bet >15% of your stack. When it's close, I'll click the pot button and see how much it adds up to. If a pot sized raise is >15% of my stack, then I'll either fold, call, or shove. If a pot sized raise is <15% of my stack, then I'll either fold, call, or 3 bet pot.

The ranges for those respective 3 bets will be quite different. Note the size of the investment relative to the existing pot and analyze the risk:reward ratio to see why.

As far as being totally committed? If I'm getting >5:1, then I'm calling any hand that I got to that spot with. In this case, even if my opponent has (AA) then I'm getting good odds to chase. However, this should rarely happen if I'm sizing my raises correctly.

You realize this means restricting yourself to a call, fold or 25x shove against a min open and 50BBs, for example? I don’t know anyone else that advocates regular use of such large 3-bet sizes.

I tend to use a similar rule, but I usually won’t 3-bet more than 1/3 of my stack without shoving.

3-betting less than pot to stay below the commitment threshold is good, I think.

It depends on the ranges and pot odds.
If someone shoves on you and the pot lays 2:1 on a call, and your hand has 33.3% equity against villain's range, it's a breakeven call (in ChipEV terms at least). If villain's jamming range is AA only and you get those same 2:1 odds, you should fold kings.
I'd recommend you literally do the maths for a few typical spots. See how much equity you need to break even if villain minraises, you make it 6bb and he jams 18bb or 20bb or whatever. (Remember to include blinds and antes for the pot odds). When you've done the pot odds math, you can compare equities of hands vs ranges to see which hands can or can't 3-bet/call off at various stack depths.

FWIW, I 4-bet/fold with over 30% of my money in the middle in cash games fairly often. I don't feel "committed", because when I make the 4-bet, I already know I'm folding to a 5-bet.

In a tournament setting, you can probably still have a 3-bet/folding range (i.e. a "bluffing range") at 15bb deep in theory, and definitely at 20bb, but in microstakes games it's inadvisable to bloat the pot pre-flop with <20bb if you don't want to play for stacks.
In short, if getting jammed on will put you in a spot where you suddenly have to do some pot odds math and you're not sure if you're getting the right price, it's often a sign that you shouldn't have 3-bet in the first place. Try to avoid situations where the pot will lay odds that mean it's mathematically correct to call, but you really don't want to (because you know you're getting stacked the majority of the time).

Thanks for your replies, I appreciate the help.