If the breaker isn't tripping (all your bedrooms are not losing power, just the light on the fan), it's not the breaker being overloaded.
How is this wired? Is the "dimmer switch" controlling just the lights on the fan, or is it the older fan's speed control? Or perhaps if I ask "How do {or did, with the old fan} you turn on the lights alone, fan alone, or both?" - is there a wall switch or speed control for the fan, and a separate wall switch for the lights, or does one wall switch (dimmer?) turn on the fan/light and then you select if the lights are on by a pull-switch on the fan? You have a remote control now - that would probably affect the way things should be wired.
Why do I ask all this? The only way I can imagine that you're causing something to overload with 120 watts of light bulb is is if you've got the fan motor attached to the dimmer, which is almost certainly not right...either the "dimmer" is a fan speed control, probably sized for the previous fan motor (did it have lights?) or it's only supposed to be attached to the lights, if it is a dimmer.
A dimmer won't work right as a fan speed control, and a fan speed control won't work right as a dimmer, and if both the motor and lights are attached to either one, it would not work.
It's good that you found this, in the worst case, this situation can lead to an electrical fire.
Check Connections
First thing to do is turn off the power, and check all the mechanical connections at the switch. If this is aluminum wiring please comment and I'll update the post, as there are other considerations if so.
Are there wires doubled up on a screw (they should not be)? Is any of the wiring loose? The wire should be wrapped around the screw or connected to the pigtails using wire connectors (depending on type of switch) like this:
Check for overloading
Calculate the total current draw on this circuit (controlled by the switch) by adding up the wattage of each bulb. This must not exceed the rating of the circuit, switch and wiring.
Most single pole switches are rated at 15A (1800W at 120V) but older ones may be less. Dimmer switches are also often less, often at 600W.
If you calculate more than 1400W, post a comment and I'll add some additional detail to consider about overall circuit loading and wire sizes.
Replace switch
If the switch is particularly old or feels loose or "mushy", simply replace it with a new one.
I'd actually replace it no matter what, since it's shown signs of overheating which could at least mean it's been damaged (even if the switch itself wasn't the initial cause).
You might also consider 'spec' or 'commercial' grade which are slightly more expensive but are much higher quality and will last much longer.
Best Answer
Unlikely that the switch isn't heavy duty enough. But the contacts in the switch and/or the connections to the wires may have problems. I suggest replacing the switch just in case it is having internal problems. When you do that, check the wires for any evidence of arcing. If the wires are not in good shape, cut off the end and strip a new section to attach to the screw terminals on the new switch. Obviously turn off the circuit at the breaker before doing any of this.