I'm assuming your full licence has been revoked under the "new drivers" regulations (six points within two years of passing your test).
There is no right of appeal against the DVLA's revocation, only against the points which activated it in the first place. The DVLA may well have discretion not to revoke, but I've never heard of it being exercised.
Unfortunately for you "No Insurance" is what is known as an "absolute" offence. This means you either have it or you don't,and it is absolutely your responsibility to ensure everything is in order. If you don't have it you will stand convicted. There is little available in the way of mitigation, so an appeal against the imposition of the points is likely to fail.
You still hold a licence but it is a provisional one which means you must take a test (both parts) again.
If you no longer intend to drive you need to contact your insurers to see if they will allow a rebate for the unexpired part of the policy.