SCRs do not normally fail open, they normally short out. SCRs are easily tested with a battery and some sort of a load (light). This is a crude way to show that the SCR turns on when a gate signal is applied and stays on when the gate signal is removed. If the SCR does not gate on I guess you could say that it failed open.
If you are talking about a power SCR there are certain test equipment that will test at the operating voltages (up to 1000V). Sencore makes a SCR tester along with a few others.
Yes it can be done on some circuits. I routinely test SCRs without removing them from their heat sinks. You will have to lift the gate lead (white wire) and the cathode reference wire (red wire). I use a six volt ordinanry flash light lantern with three test leads, two leads basically replace the lanterns on/off switch (which will now be the SCR) and the third lead goes to the positive battery terminal. You can test leakage current by the brightness of the bulb and you can turn on (gate on) the SCR and verify that it turns on. I have used the Sencore test equipment with limited success on installed SCRs. If there is a circuit path around the SCR (diode, choke, etc) the Sencore has problems.
ccauw
Depending on the protective circuit devices installed and what you are looking for, you may have all kinds of problems testing installed power SCRs. It is obviuos from your original question that you are not yet familiar with SCRs. Why don't you just tell us what is going on? Is it not working? Does it not shut off? Someone here can probably help but we need to know where you are going with this rather than guessing. Quando Omni Flunkus Moritati
