What issues should I be aware of when buying a 60's house that hasn't had anything done to it since it was built?
As much as I'd like to think I know what I'm doing when buying new properties, I ALWAYS hire a home inspector.  If you are concerned about Asbestos, you aren't going to be able to negotiate on price or to get it fixed unless you have a home inspector come in.  The added bonus of hiring an inspector is that if they miss something that should of been caught during the inspection, like termites, they are insured to cover it for you.

Further, if you suspect Asbestos, you should work testing for that into your P&amp;S contract. The tile floors and ceiling plaster are just as likely to be asbestos.

Moreover, an untouched house from that era is likely to have a leaky roof, and a leaky roof means you have mold (another reason to have an inspector).  The air ducts may / may not be full of dust. The intake plumbing may be made out of led. The cast iron waste lines may be rotting and depending on water quality you may even need to verify the that the copper piping doesn't look like it has any pitting. At this point, the the foundation likely has a few cracks and you'll wanna test for Radon if you are in an area prone to in.

I just noticed Jasper posted while I was writing this. While I agree with him, you'll likely want to update your windows. The electrical capacity probably isn't as much of an issue as it would have been a decade or two ago now that people are moving to LED bulbs and laptops. Similarly, if your appliances either work or they don't - This isn't a 60s specific thing. Moreover, I'd argue appliances were better built and easier to repair back then, so if they are in good shape may even outlast a modern one.

TLDR; Don't do it your self. get an inspection!