I used to be firmly against filling with hot water, because there are several real bad things you can do to a nice new tub with it. But most have been mentioned above, and if you are careful with it, you will be in your tub much faster.
Let me recap:
Don't use soft water. This lets a lot of people out right away.
Don't put the hot water right into the tub, add some cold first. Best to have two hoses running.
Don't let the temp in the tub get over 104 or you might set off a high-limit device. Shoot for the 90's and let the tub bring it up the rest.
Don't use the drain at the bottom of your water heater, it will blow crud into your tub.
Don't turn the tub on until it's full.
Keep the cover on if you are waiting for the water heater to reheat itself.
It is normal to fill a tub from the filter area, perhaps even through the filter standpipe - but don't do this with hot water, it can trip high-limit devices and possibly damage systems.
If you change the setting of the thermostat on your water heater to get hotter water for tub filling, be SURE you set it back to your normal setting when done, or you may have a scald accident in the shower the next day.
You know what? I think I'll just go back to saying, 'don't do it.'