This may sound silly but can I glue ceiling tiles on to/over my existing drop ceiling?

This is not going to work and a drop ceiling is not designed to be covered with anything == take those old panels down and spray paint them for a successful change of color =========

You can try but why not just remove the old tiles and replace them. As for asbestos, doubt that is present in ceiling tiles in a drop ceiling, much more likely in floor tiles.

Make sure there is no electrical junction boxes or plumbing shut offs above the drop ceiling. It is illegal to install a wall or ceiling over electrical boxes without having removable panels for access.

it might not be able to handle the weight..

HI yes you can with the correct glue.

Yep

No. Why would you?

Just remove the existing tiles and install ones you do like.

The integrity of the tiles themselves is insufficient to hold a load of any kind. Leave them alone for a few years and you can see them beginning to sag in the middle. Get them even a little bit wet and the sag can become very, very pronounced. But removing a tile is literally the work of a few seconds, and installing a new one is just as easy.

[edit]

Reading your updates, I see where you're coming from now. You want to encapsulate the material rather than remove it, in case it is asbestos.

Virtually all US manufacturers stepped away from asbestos in 1976 due to huge consumer pushback, even though ceiling tiles weren't explicitly banned at that time. If your house was built after 1976, it's probably okay.

Few things are worse than having a fear but no knowledge. Things you know about but fear anyway aren't fears themselves - they're concerns. You know what you're up against and have some idea of how to deal with them. So to help you on that score, here's some useful reading material:

http://inspectapedia.com/hazmat/Asbestos_Ceiling_T...

http://inspectapedia.com/hazmat/Asbestos_Ceiling_T...

https://www.bergmanlegal.com/ceiling-tiles-contain...

http://inspectapedia.com/hazmat/Asbestos_Ceiling_T...

If it helps you at all, I've been handling ceiling tiles off and on since 1990, some of them were at least 10 years old at the time, and I haven't been exposed at all. And if you're still concerned, there are asbestos testing kits available everywhere, and you don't even have to have the kit:

http://www.emsl.com/Page.aspx?ID=284

Good luck with it.

Can your drop ceiling carry that much extra weight? I don't think so. Most drop ceilings are a simple thin aluminum framework with VERY light panels in between the rails. Why can't you find newer, more stylish drop ceiling panels?

Response to Update: If you wear a construction/painting mask (found in most hardware stores) your asbestos exposure is minimal. And just living in a room with asbestos tiles is dangerous. Asbestos has been outlawed in the USA, for a good 50 years, at least, so the likelihood of the panels containing asbestos is minimal.

You can, but it won't work. Drop in ceiling panels will not support weight.