What happens if you draw too much current from a 12 volt sealed lead acid battery?
Well...

If your battery says it can deliver 65 amps safely for 5 seconds, that likely means that exceeding this time limit (or even further increasing the amperage) will most certainly pose a moderate to severe fire risk. Trust me, boiling acid is not fun, not even for an AP chemistry student. Always remember to have a class D fire extinguisher on hand if you ever try to do these kinds of things.

If however you decide to pull this [really stupid] stunt and your luck prevails, get ready to take cover, because your battery might be internally shorted due to the bulging of the lead plates and  beginning to enter thermal runaway: a positive feedback loop in batteries that is basically a unintentional self-destruct mechanism. However, this risk is most prevalent in flooded-type lead acid batteries. AGM and Gel batteries have different internal construction that prevent this from happening, but it still does happen on occasion due to manufacturing defects.  

If you should have a flooded lead-acid battery, that has no PRV (Pressure release valve) installed, your results might lean more on the volatile side; spraying scalding acid everywhere as the gas pressure inside the case exceeds the case's structural integrity.

However, if you're luck prevails again, and the battery in question has a PRV, your results will most likely be milder, as the PRV does it's job and vents the excess gas, preventing a catastrophic explosion and saving your beautiful face.

However, after either of these events occur, it is strongly recommended that you dispose of the battery, as it is now a serious liability, and you'll be better off just buying a new one than trying to recharge or repair (how you repair a lead acid battery, I will never know) the battery.

Edit: Fixed typos and general stupidity.