With no cab connected to your head, you'll most likely only fry your output transformer, but a competent amplifier repair shop will be able to fix you up for a few hundred or more, depending upon the power output of the amp. The more powerful the amp, the heavier the output tranny and thus the higher the price tag.
Seriously though, whoever told you it was cool to play without a speaker connected to the amp should have their head examined or split the cost of the repair bill. If you have a headphone jack that mutes the speaker when the phones are plugged in, that's OK, just be sure the cab is always connected and presenting a load to the output transformer whenever you have the amp plugged in to your power supply. Even in the case of inadvertantly turning on the power - with no guitar plugged in and no input signal present - you can do serious damage to your rig.
Depends on the amp, if it is a solid state amp, it will most likely be fine. Tube amps have to have a load. What kind of amp is it? Check the manual, it should say "do not use without a speaker load connected" or something like that if it will hurt it.
"Most" high power solid state heads have a thermal protector built into their circuitry. The minimum impedance rating should be listed on the rear panel near the speaker connection jacks. If the minimum load is, for example 4 ohms, connecting a speaker load of 2 ohms will cause the protector to open and the amp will shut itself off, thus protecting the transformer. The same goes for no speaker load - zero ohms - the thermal protector will shut the amp down until a load above the minimum is properly connected.
A dead short, i.e. a straight wire across the outputs, is 0 ohms, whereas an open, i.e. nothing connected, is infinite ohms, which is why it usually doesn't hurt ss power amps. Really, I studied this stuff for 4 years in college.
It doesn't work for tube amps though, they need a load across them.
Man do I feel like a dork! The last time I repaired my amp, I had to replace one of the input jacks. It was the 3 conductor, "shorting" type jack. I just checked the wiring schematic and the speaker outputs are the standard, "non-shorting" type.
No problem, I really wasn't trying to sound like an a$$, just didn't want anyone misled. An amp maker would never use a shorting jack on an output, maybe a switching jack that put a "dummy" load in place in case nothing was hooked up (Koch to name one). The reason for the input jack to short to ground it to kill noise when nothing is hooked to it.