How can you identify a three phase distribution board without removing the case?
MCBs in vertical columns rather than horizontal row.
If you have labels marked L1, L2, L3, would these generally be for each phase?
Yes unless board has been connected to single phase supply with 3 incoming phase terminals parelleled.
Can one of the power feeds drop out and the other 2 continue running?
Yes if the upstream protection is unlinked e.g. fuses of which one blows, or there is a disconnection in one line. If there are delta-connected loads on the board they will tend to try and backfeed the lost phase, raising it to some random voltage.
If you have L1,L2,L3 which takes up three breakers, but you install a single RCD, will this only run on the single phase its connected to?
You would not connect an RCD to an outgoing way of a DB, either single or three phase. If you meant RCBO, then yes.
Why would it only be something like a motor that would trip the MCB but not a heater?
The electrical load of a motor depends on the mechanical work demanded from it. Normally, that load is shared equally between the three windings and therefore the supply phases. If one phase is lost while it is running the entire load will be transferred to the one winding* between the two remaining phases, increasing the current dramatically. If the motor is on anything more than light load, they will become overloaded above normal flc and may trip the overload relay or MCB.
The electrical load of a heater depends only on the voltage you feed it. If the supply fails to one phase* of the element bank, that third goes cold and the others carry on as normal, without overload.
*In both the above cases there are slight differences according to whether the load is star or delta connected and whether the load star point is connected to supply neutral, but the principle is unchanged.
In fact, with the motor, the overload might be worse than you expect because it will attempt to regenerate the missing phase and backfeed it to the DB at full voltage. As a simplified example, if you have 3kW of motor load and 3kW of balanced heater load on the DB, the motor MCB will normally see a load of 1kW on each phase. Knock out one phase of the supply and the motor will try to draw enough current between the other two to power not only its own 3kW but also the 1kW of heater on the dead phase, plus extra losses as it will be running lopsided and inefficiently.