Wipe it down with paint thinner and see what comes off.

I'll bet your finish was laquer....and something...most likely just body oils from hands at the edge of the table has caused it to get gummy.
Do as Brick says....you will likely wipe off all the finish....but...is that bad?
Whenm you know it's clean, put another coat of finish in it.
Linda C

Try paint thinner first. If that takes off the 'gummies', you can do one more application(getting all the gummy stuff off). Then sand lightly and refinish with varnish or poly(I like varnish better as it is more repairable later and poly is not)

If the paint thinner does nothing, try alcohol---rubbing type is fine. That dissolves shellac. Same procedure.

If neither paint thinner or alcohol work, then you need to try lacquer thinner---you can get that at hardware stores, paint stores, or home improvement stores. It is NOT the same as paint thinner.(Just in case a misinformed clerk tries to tell you that)

Be cautious. If the finish is intact and not worn through, you should be able to salvage it. I have seen many pieces that looked as though the finish was ruined but was just a build up of wax and dirt etc. Follow the suggestions above to clean the buildup with soap and water and mineral spirits, then use 4/0 steel wool and rub hard and straight with the grain. If the finish is still intact this should revive it. Follow this with Guardsman polish, the kind that is thinned with water has worked best for me. Apply the polish and rub it all off with a soft rag with the grain.
Don't use lacquer thinner on lacquer unless you know you want to remove the finish, because that is what it will do.
I have gone on several service calls where they were sure that the finish was ruined/defective, only to rub the build up off with nothing but 4/0 steel wool, amazing the client.