Can I get compensated for funiture damage due to bedbug infestation? Can I take this issue to court?

I am lawyer in CT, not NY, but I have dealt with similar issues here and the law is not increadibly different since NY and CT generally follow the same rules.  You are in a situation that many people are facing in NYC and other places.  In fact, my sister in Italy had an issue with this when she stayed in a hostile.  With respect to your situation, you need to place the landlord on notice by writing it a letter.  The landlord should make arrangements to hire an exterminator so that you do not have to pay for it.  I would also check your lease to see if it talks about this sort of situation.  Bedbugs are not usually in a lease but you never know.  I would write the landlord a nice letter asking him to compensate you maybe in the form of free rent so you can allocate that money to replace the furniture.  If the landlord is non-responsive, I would file a small claims suit (or regualar suit in the superior court against him for breach of quiet enjoyment of your premises and negligence and nuisance (you rather not try to use your renters insurance as it may not cover bedbugs and your premium may go up).  Hopefully the suit gets the landlords attention to deal with you on this.  If you rather not sue the landlord, then you should look into a renter's insurance claim and pay whatever deductible you have.  Make sure you take your son to the Dr so that he can get treated.