How deep to consider folding a full house?

This is effectively a 'how do I play poker?' question because its so broad. You think it's not because you specified the board and stack depth. But without preflop action, reads, post-flop action, the actual suits of the cards involved etc. etc. it's impossible to say.

Like if you're playing $800 effective with the whole table at $1/2, you flatted in the BB with J6, 5 ways to the flop, checks to the river with this $10 pot and then suddenly two of the biggest nits that every nitted a nit shove all in ahead of you, sure just laugh and fold because you lose to one or both of them.

Conversely say you're $800 effective at $1/$2 and villain opens the BTN to $20. Reads:

1. This is his usual raise size.
2. He opens the BTN 100% of the time if folded or limped to based on many hours of play.
3. He has folded to your 3bet 4 times out of 4.
4. He relentless attacks weakness postflop.

You 3bet from BB to $60 with J6ss. Villain calls.

Pot is $121. Board J♥J♣6♥

x, villain bets $80, call.

Pot is $281. Board J♥J♣6♥ - T♥

x, villain bets $140, call.

Pot is $561. Board J♥J♣6♥ - T♥ - 2♣

x, villain shoves $520.

Pretty easy call.

A game of incomplete information, but you find clues and make a decision. Most of the time, when you flop a full house you’re good.

I think what you’re asking is at what point is the money pressure deep enough to make you fold a full house.

Can’t answer without specifics
In your example, I would risk any amount. You dominate the board and not many hands villain can have. This would be more about manipulating a call from hands that can’t have much.

As a conservative player, this does come up when villain could have a lot of hands that beat us. But change your mindset, you don’t ever feel bad/scared in these situations.

Evaluate the info (become good at collecting it), make a decision and don’t look back.

Tip
Be careful about leveling yourself with a very strong hand at low limits. More than a few times all the evidence will point to villain having a stronger hand - and he won’t have it.

What is relative hand strength.

In the example you gave (J66T2) then you have the 3rd nuts, with a very unlikely combo. ... so if lots of money tries to go in you need to work out how likely JJ/TT (only 4 combos.) are vs. T6/62/22/flushes/A6/6x/JT/etc.

Or even V misreading the board.

FWIW with that particular hand I probably wouldn't be worried until we went over 550bb each.

Although I wouldn't be in the hand if V raised and seems likely to have JJ/TT preflop, and if the SPR was high with 400bb each I probably wouldn't be assuming we can get it all in without some reads.

You should always "consider" all your options at all stack sizes. If you're 1000 bbs deep, then you will probably consider most big decisions longer.

Should you fold the second nuts sometimes? Yes. Consider Polk's fold of 2nd nuts to Helmuth on HSP. Absolutely the right decision. Being deeper provides more opportunities to make big folds because there is more information, whereas when you are shallow the number of hands willing to pile in is larger and the alternative bet sizes are more limited.

In short, if you can determine at 100bb that V has a better hand with a high enough certainty that calling is -ev, you should fold (even the second nuts). But situations where you can have enough certainty to make that fold will be more frequent the deeper you get.

Echoing what WereBeer said - you're asking a "how to play poker" question.

I can think of two times I folded a FH off the top of my head. One was a multi-way pot situation where it was completely obvious I was beat. I got to see the showdown, and I was right - I was beat. That was maybe 2-3 years ago.

The other time was recently. I flopped trips with AT on TT4. Action checked around. I led out on a turn K, and got one caller. The river was another K, and I check-folded to a not-all-that-large bet from a guy who seemed kind of nitty, and who was playing off a shorter stack.

I mucked face up. He didn't show. I assume he had a K. I was targeting a K when I bet for value on the turn. It seemed logically consistent to fold river on another K. He refused to tell me if I made a good fold, even later.

I may have folded a FH other times, and just don't remember. The fact that I can remember all the details of those two hands suggests a level of emotional scarring from having to fold a big hand, and leads me to believe I probably haven't folded very many FH's.

All that said - the deeper we are, and the bigger the bet we're facing, or the more hot-and-heavy the action is, the easier it would be to find a fold. I guess it's all about the SPR.

Side-note - I'm sure I've lost more money calling big bets with flushes than I've lost calling big bets with FH's.