Are we talking about a phone that is connected to some type of an alarm system or about a phone that a person would pick up and it would auto-dial 911?
If it is the first then most 911 centers would not permit that, but unless the system is making a lot of false calls they would never know. Some centers have a special number for alarm systems to call in on, but most alarm system go to a central station monitoring service and someone from the monitoring service would call the 911 center. I am not a fan of central station monitoring because while most monitoring centers do much better than what the NFPA standard requires, the NFPA standard permits up to a 4 minute delay on "re-transmitting" the alarm to the 911 center.
While we can't prohibit the use of central station monitoring, our 911 center strongly encourages a direct connection to our center via a radio system. We do prohibit alarm systems from calling in on our 911 lines, but we have no way of knowing if a system is set up to do that until we start getting false calls from the alarm system.
If the phone in question is a phone that is physically used by a person our rules would not prohibit that phone from being an auto-dial phone. As long as that phone is not in a location where the handset would be getting knocked off the base and the phone would be calling 911, I don't see an issue with using an auto-dial phone.