A database server is a database server; regardless of where it's sitting. It's fairly easy to just repoint a line of code to a new database server.
As with any technology related investments, you have to consider the following:
- latency; how much tolerance can your app or your company manage?
- performance; will the replacement solution provide enough horsepower to do all your queries in a timely fashion?
- capacity; will the replacement solution provide enough storage for growth?
- support; service level agreements are very important when it comes to who is responsible for network/systems issues and the mandatory response times. What are the terms of support?
- high availability; does the provider have HA capabilities in place? If so, what is their framework/infrastructure for this?
- fault tolerance; does the provider have FT capabilities in place? If so, what is their framework/infrastructure for this?
- load balancing; is this a dedicated system/resource or shared? How will the provider deal with performance requirements?
- security; is this a dedicated system/resource or shared? How will the provider manage risks?
- cost; are you getting a good deal for all of the items above?
That being said, I'd think your biggest issue would be latency, depending on where the app and database servers are physically located, as well as the bandwidth of the network connecting the two locations.
You might be better off moving your entire environment onto a more stable provider's infrastructure. However, to be sure of which way to go, you need to do a risk and cost analysis. These will help you determine:
a) if it's technically feasible
b) if it's financially reasonable
c) if it provides pros that outweigh the cons
d) if it resolves your original concerns