The short answer is no; a telescope will not damage your eyes. But there is a caveat to this which we will come onto in just a second, as we need a little background on what telescopes actually do.
The first thing to note is what telescopes don't do. Contrary to popular belief, telescopes don't make things brighter.
All they do is magnify the dimensions of the object you are looking at, which gives off the effect that the object is brighter when in reality it's just magnified.
The higher the magnification can go on your telescope, the surface is broader, meaning that the light is no longer focused on one point but distributed over a larger surface area.
With that information in hand, there are some instances where looking through a telescope can damage your eyes.