Can you keep playing when you're out of time?

Hi, I just wondered if it during a tournament is allowed to keep playing when you're out of time and your opponent doesn't notice.

Also, do you lose if you make an illegal move?

I would think so....

Most modern clocks beep or flash (or both) when someone runs out of time, so it's very hard to not notice.

But yes, if your opponent doesn't call it, then you can keep playing.

If you make an illegal move, then nothing happens... unless your opponent calls you on it. The penalty is usually a clock penalty (you lose a few minutes or they gain a few minutes).

If you repeatedly make illegal moves then maybe you'd be forfeited. This would be rare though... I've never heard of it.

Not 100% sure but i believe the rule for the USCF is, if your opponent doesnt say anything then keep on playing.  Its just like making an illegal move, Its up to your opponent to notice, and say something. 

Thats interesting 

From the FIDE rules:

6.8: A flag is considered to have fallen when the arbiter observes the fact or when either player has made a valid claim to that effect.

If you're lucky and nobody notices before you reach the time control, you can go on. I won't go into what happens if both flags have fallen (time ran out) and you don't know which fell first.

7.5: 

1. If during a game it is found that an illegal move has been completed, the position immediately before the irregularity shall be reinstated. If the position immediately before the irregularity cannot be determined, the game shall continue from the last identifiable position prior to the irregularity. Articles 4.3 and 4.7 apply to the move replacing the illegal move. The game shall then continue from this reinstated position. 
   If the player has moved a pawn to the furthest distant rank, pressed the clock, but not replaced the pawn with a new piece, the move is illegal. The pawn shall be replaced by a queen of the same colour as the pawn.
2. After the action taken under Article 7.5.a, for the first completed illegal move by a player the arbiter shall give two minutes extra time to his opponent; for the second completed illegal move by the same player the arbiter shall declare the game lost by this player. However, the game is drawn if the position is such that the opponent cannot checkmate the player’s king by any possible series of legal moves.

I'll summarize the relevant parts of appendix A: if rapidplay (fixed time (or time plus increment times 60) of 10-60 minutes) or blitz (faster) apply and there is insufficient supervision the arbiter or opponent can claim a win (or a draw with insufficient mating material) if the illegal move is noticed before the next move is made.

There are no beeping on our clocks, and the flashing is usually a black flag or a small triangle. I was wondering for blitz mostly. Often when I run out of time, the other guy sits there for like 10 secs before I say: "Dude, I lost on time". But now I'll stop doing that.

I guess I can claim draw if both run out of time?

For blitz, it looks like it would be a draw.

In my chessclub clocks make sounds when somebody is out of time, so for me no

Cool, so in a long chess game where I touch my queen, but realise any move with it would be a serious blunder, I can just make an illegal move with it and get the position reinstated? (with -2 min, ofc)

Touch move (4.3) still applies after making an illegal move.

Yeah, but I was talking about it as a way to avoid touch move. You have to move, so you just make an illegal one to get back to previous position.

No. If you make an illegal move, the position reset and you must make a legal move(if there is one) with the piece you originally touched.