You aren't going to damage the power supply, but you might damage the strobes connected to it as Shawn mentioned.
An unregulated power supply splits the wattage among all of the connected heads based on resistance. A lower resistance light will get more wattage and therefore be brighter. Putting a low resistance strobe (e.g. hide-a-way) and a high resistance strobe (e.g. linear 900) together on an unregulated power supply will cause the hide-a-way to be very bright and have a shorter life and the linear strobe to be very dim.
A regulated power supply supplies a fixed wattage per head to all heads regardless of resistance. Regulated power supplies can either be regulated per head or per pair of heads. So you can have a 90 Watt 6 Head Regulated Supply that:
1) Delivers 15 Watts Per Output regardless of the number of connected outlets
or
2) Delivers 30 Watts Per PAIR of Outputs regardless of the number of connected outlets.
In this second situation you effectively a multiple unregulated power supplies within a larger regulated power supply. You can still have the problems listed for an unregulated power supply with this type of setup, but since most strobes are installed in pairs most problems can be avoided by installing the same type strobe on paired outputs so that that 30 Watts is split among a two strobes that have the same resistance. However, this isn't always as simple as it sounds, sometimes they are paired 1&2, 3&4, 5&6, sometimes 1&3, 2&4, 5&6, etc. You really have to take a close look at the manual.
As if things weren't complicated enough, there are what I will call Smart Power Supplies. These power supplies have the ability to turn down power based on having an open circuit (burnt out or disconnected strobe tube).
Lets say you have a 90 Watt 4 Outlet Supply that is regulated by PAIR to 45 Watts. If you have two HAWs connected each getting 22.5 Watts you are fine, but as soon as one blows out the other is getting the full 45 Watts of that output and you are going to have two blown out strobe tubes. To counter this problem, the Smart Supply Knocks the wattage down to 30 for that PAIR of outlets if only one is connected. This isn't regulated because the output wattage isn't fixed per outlet.
I hope this answers more questions than it causes.
-Jared