Check the boiler water pressure. Generally, it needs to be something like 14-20#. If it is low, the boiler will shut down. You should have the owner's manual (if you don't, check online for it). It will have some troubleshooting info.
There are various interlocks, and if you can figure out the wiring diagram, it is generally fairly easy to see what switch or control is not functioning. To do this, you'd need a meter and know how to use it (well, you could use a 24v lamp and some leads, but that is a little harder). Various interlocks might be things like: flame sensor, flue damper, pressure (vacuum) indicating there is a fan on, low water pressure, high temp and maybe others. If the circulator isn't running, the boiler can get too hot since the water isn't moving. Lots of things, but if you are logical, easy enough.