You do not want to put a 250 watt bulb into a socket that only supports a max of 100 watts. No way. Don't do it.
The bulbs that come with this lamp at 5000K in color temperature. You will not like that in your home. It is a terrible cold and harsh type of light. You will feel like you are in a sterile hospital operating room. Even a 4000K bulb is "too white" for most people in home lighting. 4000K is OK in a garage or a kitchen, but probably not in your den. You're mostly likely going to want 3000K or below, the most commonly found being 2700K, which is very similar to incandescent bulbs. But this is just the bulb that comes in the lamp, which you can change. Expect to want to do that immediately if they give you a 5000K bulb.
Note that traditional 3-way bulbs have multiple filaments on the inside. The different light levels are implemented by turning on one filament, or the other, or both. That's why you need a special 3-way lamp to use 3-way bulbs in. This is all technology from back in the incandescent days.
With LED bulbs (which is what are supplied with the lamp), you do not have distinct/discreet lighting levels like the old 3-way incandescent bulbs did. You have a continuously dimmable level of light. The lamp simply includes a dimmer built in. In the case of the specific lamp you are looking at, it indeed talks about "three light levels". If this is true, they would have intentionally designed this limitation into their dimmer circuit. Or maybe they're really stupid and indeed are simply decreasing the voltage like the old dimmers used to do. This would be stupid because in general, LED bulbs do not like this one bit. They don't function well. Maybe they found some LED bulbs that tolerate this better than others, and those are the LED bulbs that they include with the lamp. I can't believe any company would actually be dumb enough to do that. Did this lamp come from China by any chance?
The way old incandescent bulbs were dimmed is by reducing the voltage. The bulbs are fed the typical 120 volts that come from your home wiring for full brightness. The old incandescent dimmers would cut that voltage down to lower levels to give you the dimming feature. You could do that with a simple rheostat, but more modern dimmers did it electronically.
With LED bulbs, you do not dim them by lowering the voltage. You dim them by cutting off some of the sine wave cycle. So instead of being a nicely rounded waveform, part of it is abruptly cut off in a straight line. On top of this, many electronic dimmers worked by cutting off the leading edge of the sine wave, but LEDs like it better if you cut off the trailing edge. They might still work with a leading edge dimmer, but they work better with a trailing edge dimmer.
What happens if you use the wrong kind of dimmer for your type of light bulb? Generally, the bulb will flicker annoyingly. Sometimes it won't dim, but just turns off when you try to dim it. Sometimes the bulb can overheat. Sometimes the bulb will develop an off-color cast over time. Sometimes it will lose brightness over time. Usually, the bulb will not last anywhere near as long as it should. The bulb will often give off spurious noise in the radio frequency spectrum. If someone in your house is a radio operator, they will probably hate you for that. Bottom line: using the wrong kind of dimmer for your bulb type is probably not going to blow up your house on the spot. But your results will most likely be far from pleasing. If you never use the dimmer and always leave it set for full brightness you're good to go with whatever bulb type you want to put in there. But then you have to wonder why you bought a dimmer in the first place. Note that if you use an incandescent bulb with an LED dimmer, it will most likely dim just fine. You'll probably even be happy with the result at first. Until you have to replace the dead bulb two weeks later. You may get some flicker during the bulbs shortened lifespan as well.
Putting a 250 watt bulb in a 100 watt socket is a fire hazard. A pretty big fire hazard. As I said above, don't do it. UNLESS you are talking about a "250 watt equivalent". And LED bulb of maybe 40 or 50 watts would be somewhere near a "250 watt incandescent equivalent". I can't say that I've ever seen a bulb like that at the Home Depot, but somebody, somewhere probably makes one. I doubt that's what you have on-hand though.
Back to your original question: Do I have to use LED bulbs? Technically no, realistically yes. Because of the dimming. And you will probably want to replace the high color temperature LED bulbs that come with the light very quickly after you turn the lamp on and see the cold, harsh light that comes from a 5000K bulb.