Is there a word for the condition of being infested with ticks?

I don't think there is a word that refers to possessing ticks as a condition, but you could use "tick-ridden" or "tick-infested" as an adjective:

The dog is tick-ridden.

Ectoparasitism TFD is the medical term:

the state in which the ectoparasite [ticks, lice] are living on the surface of the host's body.

He went to the doctor with ticks, and was diagnosed with ectoparasitosis.

For the second sentence example you can simply use "ticks" - "don't go near him he has ticks", just as you would with lice or fleas.

If sticking to a single word is not essential you might want to be more specific and say "an infestation of ticks" - to make it clear that the subject has a large number of the parasites, not just one or two.

"an infestation of ticks" could also work in the first sentence example. "He's suffering from an infestation of ticks"

Merriam-Webster:

ticky -er/-est

: affected or infested with or full of ticks