All it seem to do is to resample the image with a stripped down version of it... thus creating a small size file.
I think there is no issues using them.
What it does is explained clearly on the first page of the website:
TinyPNG uses smart lossy compression techniques to reduce the file size of your PNG files. By selectively decreasing the number of colors in the image, fewer bytes are required to store the data. The effect is nearly invisible but it makes a very large difference in file size!
As the result is probably no longer a 24bit PNG, you may face issues (in libGDX for example) but any problem with the image will be immediately noticeable and you will have just to revert to the original file. Personally, all my PNG are compressed in my apps (but not with tinyPNG and with a lossless compression).