No, don't put your private IP addresses in the public DNS.
Firstly, it leaks information, although that's a relatively minor problem.
The worse problem if your MX records point to that particular host entry is that anyone that does try to send mail to it will at best get mail delivery timeouts.
Depending on the sender's mail software they may get bounces.
Even worse, if you're using RFC1918 address space (which you should, inside your network) and the sender is too, there's every chance that they'll try and deliver the mail to their own network instead.
- network has internal mail server, but no split DNS
- admin therefore puts both public and internal IP addresses in the DNS
- and MX records point to both:
@ IN MX mail.example.com
mail IN A 192.168.1.2
IN A some_public_IP
- someone seeing this might try to connect to 192.168.1.2
- best case, it bounces, because they've got no route
- but if they've also got a server using 192.168.1.2, the mail will go to the wrong place
Yes, it's a broken configuration, but I've seen this (and worse) happen.
No, it's not DNS's fault, it's just doing what it's told to.