They'd cause problems for a few generations, until their population collapsed from a lack of females.

Certain ant species can eat ticks, but as far as I'm aware there's no species feeding only off ticks, and Lyme and Encephalitis are really nasty. But it's an interesting question: suppose we can somehow erradicate all ticks, what effect would that have? How much do they matter for the food chain or on ecosystems in general?

You also have to ask the other way around: What do ticks do to their hosts? Animals such as moose or reindeer can suffer greatly from heavy tick infestations. Depending on the death rate this causes removing the ticks could have similar effects to removing predators such as wolves from the ecosystem. However there are many tick species, and I doubt the impact would be that strong, but who knows.

It is a part of the possum diet. They eat upwards of 5k per year.

Ground birds as well. Chicken's (and Guinea Fowl) will eat large amounts of ticks if present.