If a billboard lease on the side of a building is not filed with the county and a new owner did not sign anything to obligate them to take over the lease, is the new owner obligated to honor the lease? Can the new owner request the sign be removed and/or use for trespass?

Unless a lease says otherwise, it carries over to a new owner. Filing with any gov't agency is not relevant.

Unless a lease says otherwise, it carries over to a new owner. Filing with any gov't agency is not relevant.

Let me get this straight: you are saying that if I buy a building which happens to have a sign on it, and later somebody comes along saying "that's my sign; I am leasing the side of your building; here's your rent check.", I am obligated to honor such a lease?

I don't think so.

Perhaps a billboard is different from a lease of the building itself, but I can't find any reference to a difference on any legal site.

I would say the difference is this:

If I buy a house which has someone living in it, I am on inquiry notice that there may be someone leasing it. If I ask the seller, and he denies a lease, I still should ask the tenant, I guess.

If I buy a house which has a sign on it, who am I supposed to ask, assuming the seller denies any sign lease?

The billboard renter put start up costs into the billboard, presumably. He has a right to expect the lease to continue to it's end, just as any lessee does. If the seller lied to the buyer, the buyer sues the seller, and the lessee sues the buyer. I would say.

The seller would have a obligation to disclose any contracts or obligations with a building.

So I do believe that a buyer will be obligated to honor it, since it is attached to the building, and/or painted on the building. That would or should have been a buyer question also. Billboard income is a large business and income often large.

If for example, the building has leased equipment, such as security alarms, in it, merely buying the building, does not make that equipment yours,

In this case, I believe that the lease has to be upheld on face value.

But the issue here is, permits, if the sign is not legal, the building owner can be liable for fines and code and zoning issues. So there can not be a contract for a illegal activity. If there sign was not installed with the correct permits, and the sign is not legal to be there. That would void the lease on grounds of illegal activity.