My BF did the same thing to his cell phone with the cig lighter charger in his pickup, We took it into the phone store where he bought the phone and a 2 year contract. If that model of cell phone hadn't been obsolete all he would had to do was buy a replacement battery ,He wound up having to buy a newer model phone and the woman behind the counter spent several minutes explaining to him about over charging and the effect it has on the life of the battery. as a matter of fact she said that periodically to allow the battery to run all the way down to dead and recharge it ,helps lengthen the life of it.
Ignore the bits that don't apply to the question, I just prefer to take a run at it from a suitable distance.
Some batteries are designed to be single-use, some are designed to be rechargeable. They have different chemical constituents as well as different internal design.
Early rechargeable batteries used Nickel and Cadmium electrodes. They had a "memory effect" which meant that if they discharged from full to x% and then recharged, x% was as low as they'd get subsequently before they said they were empty. That only happened on Nickel Cadmium batteries. Sometimes they're called NiCad. I'd be surprised if there were many still in use.
More recent batteries are, for example, Nickel Hydride (Nickel Metal Hydride, NiMH) or Lithium Ion or Lithium Polymer. None of those have a memory effect.
Every one of these batteries can be destroyed by overcharging.
A good recharger has a stop-charging circuit that detects when the battery's become fully charged and stops pumping more electricity at it. (Pumping's an appropriate word, the charging circuit strokes the battery into an excited state by thrusting electricity into it rhythmically until the battery can take no more). Anyhow, a BAD recharger doesn't have stop circuitry and destroys the chemical constituents if the charging process is left on longer than the battery can absorb the extra power.
If your phone handset gets plugged into the mains to do the recharging, and it's overcharged the battery, then I'd call that a pretty useless phone handset because it has no built-in protection. Cheapo handset. Eww.
If you buy a new handset, check if it has overcharge protection and if it's got a NiMH/Lithium rechargeable battery. Don't buy it if it doesn't have both.