Hi Heidi! It depends on what you're trying to say.
If you are telling someone that you don't have a phone number (cause they ask you for your phone number), the first one is correct -- "I don't have a phone number".
If someone asked you for all the phone numbers you own, or all the phone numbers you could possibly give them (from your phone/notes), the second one works well -- "I don't have any phone numbers".
Heidi I hope it helps!