1. Don't hang up the phone.
2. Don't hang up on me.
3. Hold on the phone.
4. Hold on the telephone.
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Are they all grammatical?
1 and 2 are fine, but 3 and 4 don't work.
You might rephrase 3 and 4 to mean that you need to check on something and the caller shouldn't hang up.
Please hold while I find that information.
Please hold while I look for Mr. Jones.