As the curious one and Martin Beckett points out in the comments, the idea of trying to transmit an image in a fiber is wildly impractical and there are probably simpler solutions.
However, to answer the title question, yes most fibers have important power limits. For example, we avoid putting more than 300mW into the single-mode fibers we use in my lab. This might seem small, but keep in mind that the mode-field diameter of such a fiber is quite small, so that 300mW has been focused to a diameter of about 5 $\mu$m. There are two main limitations to this - damaging the fiber itself at the air to glass interface and damaging a connector or termination component. For example, many fibers are terminated in a ceramic ferrule, and the fiber is held in the ferrule using epoxy. If the power is too high, the laser light can vaporize the epoxy causing an epoxy reside to coat the fiber, thus increasing scattering and heating, creating a small positive feedback loop and probably causing major damage to the fiber.
We have some higher power fibers which don't have the ceramic cladding or have an "end-cap" design where the air-to-glass transition happens at a much larger mode-field diameter before the mode is shrunk to the single-mode size. But eventually these types of issues would still potentially melt part of the fiber connector. Multi-mode fibers will have much higher limits because the fiber cores are much larger. The fiber core itself can handle much higher powers which allows applications like that mentioned by Martin, but to handle such high powers extreme care must be taken. I think for example the powers Martin is referring to are coming from fiber lasers, so the light is actually generated inside the fiber and is not so subject to the issues I'm mentioning.
In the end you have to ask the manufacturer what powers they recommend / have tested for a given fiber. For example, thorlabs has a nice guide for this: