The short answer is No, they won't kill a tree. But established mature climbing hydrangea is a large plant,sometimes growing to 50' if not pruned, so you would need to keep in mind the size and strength of the tree on which it is climbing. If you are in an area where summer storms with wind/rain are common, you might want to consider the added weight of the vine especially when wet and how that might affect a small trees resistance to wind.

I've seen a very pretty example growing on a mature doug fir - no harm, no foul there and the vine is well out of reach of pruners.

Why do you ask? Do you have a problem with a tree & a climbing hydrangea now? Or are you considering a CH?

luis pr - i have a well established CH that is growing on a well established hemlock - but the hemlock has died and i don't know why.....

There can be some problems with hemlock (tsuga) in landscapes - needs evenly moist, acidic, well-drained soils, partial shade, all conditions which allow it to thrive here. It doesn't tolerate extended periods of dry any better than it does soggy soil.

As for pests, woolly adelgid will kill hemlock, but it isn't hard to spot should your tree be suffering from those:

Here is a link that might be useful: hemlock woolly adelgid