1) Is there such a thing as experience or skills being too old?
Yes, not only are some skill sets too old but they can date you as well. I am a whiz on an IBM keypunch machine but that is not going on my resume. That might make me "overqualified" for the job I am applying to.
2) If there is no such thing as experience or skills being too old, if I leave that job off my resume because it was ten years ago, will it look like I'm lying if I list those skills in a summary?
If the skill is still relevant, you can "move it up" in the resume on in the Summary of Qualifications section. You don't necessarily have to tie it in with a particular employer.