iMessage keeps asking for sign in

Hi

You may previously have enabled an app-specific password for iMessage and FaceTime, which is different to your main Apple ID password. If so, follow the instructions here:

Using app-specific passwords - Apple Support

It may otherwise help to follow the steps here, including turning iMessage off / on again and contacting Apple Support or making a Genius Bar reservation if necessary:

If you get an error when trying to activate iMessage or FaceTime - Apple Support

Are you connected to WiFi or 3G? Sometimes, when a device is not connected to a network for a long time, it will continually ask for logins to internet enabled apps.