Do solar panels burn thru batteries?

When a proper solar charger wakes up, it will take the batteries up to the absorption voltage, about 14.4V. It should then drop back to float voltage, about 13.2 for flooded and 13.8 for AGM. This is where cheap and more elaborate solar controllers differ: a cheap one will hold the battery at the 14.4V absorption voltage for a time period (typically 2 hours) while a better one will drop it back as soon as the charge current has dropped below a set threshold. On fully charged batteries this may happen in only a few minutes. The former strategy is a little harder on the batteries than the latter. Neither is as bad as letting them slowly discharge in storage.

Another detail is the correct float voltage varies quite a bit with temperature. Since solar is charging all the time, temperature correction becomes more important. The more elaborate solar (and line) chargers will have a temp sense on the battery and correct accordingly.