You'll be fine. You won't see an iguana or other reptile unless you go looking for one and put in some effort. At least that's true in Playa del Carmen proper - all bets are off if you venture beyond the town.
I've been traveling to, or more recently though, Playa del Carmen many times going back to the 1980s and as recently as about 8 years ago. I don't think I've ever seen an iguana or other live reptile there, even long ago when all the streets were unpaved, sandy tracks, and there was just a single telephone in the town. Nowadays, it's a crowded mega-developed tourist magnet, highly paved-over (no more sand streets) and someplace that I generally try to avoid (not because of iguanas, it's just seriously over-touristed).
Back in the day, I'm sure there were plenty of iguanas across the highway, as that was all just undeveloped jungle. But Playa del Carmen itself is now a HUGE, developed metropolis (by some measures, Playa del Carmen is the fastest growing city in North America, which says a lot about the mind-blowing explosion of tourism development there). I think you would have to work hard to find an iguana wandering around anywhere in the town or on the beach. Now, to be honest, it's probably been 15-20 years since I last "hung out" on the beach there for more than a couple minutes (just to stick my toes in the water on my way to/from the ferry) but I don't recall EVER seeing iguanas loose on the beach there (elsewhere in Mexico, yes, but not there). Unless someone has an iguana on a leash and is trying to get 5 bucks from tourists to take their picture with one, I don't think you'll see any. If you happen to spot one at a distance, just keep away from it (sounds like that shouldn't be a problem for you...). They are actually a bit skittish and really have no interest in approaching you. If it's any consolation, they generally move quite slowly - you could outrun them by just walking slowly (they can scoot along faster if they're very threatened, but they rarely move fast, they conserve energy by moving slow).
If you stick to the town itself (and to the beach, which is lovely, though too crowded for me), I really think you will be fine and will not see any iguanas or similar critters AT ALL. If I were you I'd try to cross this off my list of things to worry about.
When is your trip? I'll be there in mid-April and I can take a quick look around to confirm there are no wandering free-range iguanas there and will report back.