If the fielding team notices that one of the batsmen on the batting team is having a bad day, will they give up a chance at taking a wicket in order to keep him on the field? For example, if the batsman has a terrible strike rate and then he pops the ball up and it could be an easy out, will the fielder purposely not catch the ball in order to keep him on the field and continue being a liability to the batting team, especially if the batsmen ahead in the order are more dangerous?
I've never heard of such a situation occurring, mostly because I don't think it would be particularly effective. Assuming a batsman is swinging hard, the batsmen are going to get at least a run a ball if the fielders aren't trying to get them out, and probably more than that. That's a recipe for losing a game, even a Twenty20 game.
A couple of notes on the rules which you may not be aware of: there's no need to do anything like not catching the ball, just don't appeal. However, the batsman can always leave his wicket without an appeal being made (explicitly allowed by Law 27.1) or in the event that the fielding team does go so far as to not catch the ball, just retire out as allowed by Law 2.9(b).
There are two incidents that almost fit the case you ask , both involving the same batsman, Geoffrey Boycott, who was a great test mach opener, famed for his defensive qualities, but sometimes unwilling or incapable of stepping up the scoring rate when the situation demanded it.
During the 1979 World Cup final, Boycott was dropped by the West Indian captain:
Boycott, who had taken 17 overs to reach double figures, was facing shortly after tea when he came down the pitch to Richards and mistimed his stroke. The ball arced gently to Clive Lloyd, an outstanding fielder, even though he had already dropped Brearley, who was at wide mid-on. Just as slowly, Lloyd dropped it, ending up on his back with the ball on the turf close by.
Lloyd denied dropping it intentionally, but admitted that it would have been a good idea:
I could have watched them all day because I knew every over they batted was another nail in their coffin. A lot of people suggested I put [the catch] down purposefully just to keep him in... not true, but it wouldn't have been a bad tactic.
There is a video of the incident, so you can judge for yourself. I would tend to trust Lloyd's words, and you can see that he does have jump and extend to try to make the catch.
A separate incident is the other side of the same coin. If the fielding team want to keep you in, then your teammates probably want you out. During a Test match against New Zealand, vice captain Bob Willis gave Ian Botham instructions to run Boycott out in order to increase the scoring rate to set New Zealand a 4th innings total for the final day.
As per the rules it is fine, no one can force bowling team to take wickets if they no intend to. Sometimes in test matches, bowling team avoid nightwatchman wicket even any player wicket to get some time pass as per the match condition that time.
But, if a batsman is bolwed, catch or even LBW, it is umpire responsiblity to decide out even bowling team dont appeal and bowling team can never ask to umpire for a dead or no ball.