I don't think he should be dropped from the team, purely because we don't have anyone better. He makes some very, very questionable decisions regarding his personal life, and generally seems like a bit of a weasel of a fellow, but he's a good defender. We need him even more if Rio's injury problems don't pick up, which they're showing no sign of doing.
I do think he should lose the captaincy, though. For me, especially at international level, there's a big step up from merely being in the team, and leading the team into a World Cup final. Terry, while he might still have allies in the camp, is now in no position to be the general on the field. Either give it to a fully-fit Rio, or Rooney.
Rooney would be a good replacement, but my only fear would be that he has enough on his plate leading the line for England and being our main goal threat without also leading the team.
I wouldn't worry about it being too much for him psychologically, I'm fairly certain he could handle anything, but I'd worry about what it could do to his performances. If he's leading the team as well as the forward line, he might be tempted to break the discipline that has served him so well this season and go back to charging around after ever loose ball. An undisciplined Rooney is still the best player we have, but we need him in proper goal scoring form, and the captaincy could distract him from that.