Once again I seem to be asking about older UPS units. In this case we have some divisions who have put off battery replacements for a couple of years. That is to say, the maintenance tech visited the sites, found the batteries to be more than 5 -7 years old, recommended replacement each of the past two years, and now we have a tight budget year which is their excuse for not purchasing new batteries this year. The units are all ferro-resonant UPS units up to 12 years old. We had one unit fail earlier this year due to a thermal issue with the batteries. My question, to finally get to the point, is: Is there a physical danger to continually charging and using the old batteries?  

Depends upon the battery type. They come in a few flavours depending upon the technology that's used to make them. Ni-Cads, Cells, Plante cells and VRLA to name but a  few.
I'll assume you're using VRLAs. These 'usually' come in two varieties. The 5 year life, (3-5 in practice), and the 10 year life, (7-10) in practice). But these figures are just guidelines because the battery manufacturer does not know how hot the environment is, how much ripple is on the charger, how many charge/discharge cycles they are subjected to , float voltage and and quite a few other things.
Now there will be circumstances where a battery has lasted far better than expected even under adverse conditions, but these are the exception. Remember it might only take one cell to fail and the entire battery may as well not be there.
Nobody really knows or could predict whether your battery will do what you need it to with absolute certainty but I say this. I don't believe you can replenish a gel/sealed battery, and once they start to dry out they get hot. It is by no means a daily occurrence but I have seen a few battery fires,  they are not good and they were NEVER on newer batteries - always for installations where we had repeatedly told the client to change them.
You have to balance the cost to your business should you either lose the load or have a fire, against the cost of battery replacement.
Only you can make that call.
I would get them changed.