I don't know exactly what you mean by "case you".
Any employee or employer can file a complaint with the Ministry of Labour (MOL) in the UAE, and they might or might not open a case for the complaint. I guess this is what you mean.
In your situation, I don't think the MOL will make a case against you for not taking up employment with a new employer if you have signed a contract but processing for a residence visa or work permit hasn't started yet. The employer could file a civil claim against you through the court system but in most cases this would be unlikely given that the expenses of doing so usually exceed any likely benefit to them.
However I am not connected to your new employer, the MOL, or the Dubai Court system, so I can't say for sure what they will or will not do.
A threat is not the same as an action. You have to do your best to figure out how likely it is your new employer will follow through on their threat. And aside from that, an important question to ask yourself is do you really want to work for an employer who makes threats like that? To me it's a sign of future problems.
With respect to your current employer, I don't know of any reason why they cannot cancel your visa. Sounds to me more like they don't want to, either to keep you or to make life difficult for you.
The normal procedure is that the sponsor (your employer in this situation) must do the visa cancellation procedure with the MOL and the immigration department. It is not something you can do on your own.
If your contract is unlimited then the standard notice period for resignation according to the labour law is 30 days. If your contract says something different then check with the MOL or a lawyer if it has any legal validity. If you give the correct notice period and file your resignation according to the correct procedure (letter, fax, or email), then that is all that is required on your side for the resignation to be valid. Whether the employer says they accept it or not is irrelevant. A resignation is your choice and your decision, it is not a request or negotiation, simply a notification.
You might or might not receive a labour ban depending on several factors. The most common being how long you worked for your current employer. If more than 2 years you should not receive a ban. If between 1-2 years then an automatic 6 month ban is normally applied but can be lifted. If less than 1 year then you might have a longer ban or one that is difficult to lift. I'm not sure exactly - check with the MOL.
An NOC doesn't usually make any difference to whether or not you get a ban, or whether or not your visa can be cancelled, or whether or not you can get a job with a new employer.
I would suggest you phone the MOL with your question and see what they say.
http://www.dubaifaqs.com/labour-law-uae.phphttp://www.dubaifaqs.com/ministry-of-labour-uae.phphttp://www.dubaifaqs.com/visa-cancellation-dubai.php