You should absolutely not try exceeding your monitor's scan rates if it's a fixed-frequency type. You can smoke your hardware doing this! There are potentially subtler problems with overdriving a multisync monitor which you should be aware of.
Having a pixel clock higher than the monitor's maximum bandwidth is rather harmless, in contrast. (Note: the theoretical limit of discernible features is reached when the pixel clock reaches double the monitor's bandwidth. This is a straightforward application of Nyquist's Theorem: consider the pixels as a spatially distributed series of samples of the drive signals and you'll see why.)
It's exceeding the rated maximum sync frequencies that's problematic. Some modern monitors might have protection circuitry that shuts the monitor down at dangerous scan rates, but don't rely on it. In particular there are older multisync monitors (like the Multisync II) which use just one horizontal transformer. These monitors will not have much protection against overdriving them. While you necessarily have high voltage regulation circuitry (which can be absent in fixed frequency monitors), it will not necessarily cover every conceivable frequency range, especially in cheaper models. This not only implies more wear on the circuitry, it can also cause the screen phosphors to age faster, and cause more than the specified radiation (including X-rays) to be emitted from the monitor.
Another importance of the bandwidth is that the monitor's input impedance is specified only for that range, and using higher frequencies can cause reflections probably causing minor screen interferences, and radio disturbance.
However, the basic problematic magnitude in question here is the slew rate (the steepness of the video signals) of the video output drivers, and that is usually independent of the actual pixel frequency, but (if your board manufacturer cares about such problems) related to the maximum pixel frequency of the board.
So be careful out there...