What should I do about ceiling tiles that potentially contain asbestos?
As a certified home inspector I will give you some more specific advice from a Buyer's point of view. If the inspector suspected asbestos, then this warning should be taken seriously. At this point in the house buying negotiations, it would be wise to insist the "Seller" have the material sampled by a licensed tech and tested by a certified lab with a copy of the results given to you before proceeding to a closing. If the tests do come back positive for asbestos, get a quote from a licensed asbestos remediator to remove it, and a quote to replace the ceiling with some safe material such as drywall. I would recommend asking that the total amount of both of these projects  be adjusted from the selling price.

Under no circumstances would I advise one of my customers to proceed further into the purchase until this issue is resolved. If positive for asbestos, this could be an expensive fix and definitely not a DIY project under any circumstances.