With the DC ballasts the filament is usually driven from separate winding from the transformer, so when it goes open circuit, it have no influence on the main ballast functionality.
On some ballasts, one filament is connected in the feedback path, so the ballast shut down, when the lamp is removed. With these, just try to insert the lamp in both ways, at least one will light.
A bit different ale ballasts with two large transformer like cores (one is really a transformer, the second is just an inductor). Only these (two transformer-like components of about the same size) are the types, which work in the same way as mains ballasts (CFL,...) and need both cathodes to be intact. These always use either two or four power transistor and at least one control IC.
With this style is the only one containing the real lamp-fault detection/shut down circuit (the "intelligence" of the control IC allows it), but it is not 100% sure.
If there is only one large transformer, two power transistor and one small inductor (beside few capacitors), it is most likely the Royer oscillator type. These normally instant start, or at least one filament should be enough as a RS-style starting aid. If the filaments are heated, their supply comne from extra winding, so if left open, nothing happen (only the lamp is not heated). This ballast style quite well tolerate the missing lamp, so frequently the "missing lamp lockout" (checking the filament presence) is not used. But when the lamp have higher drop than designed for, the input stage is overloaded by high current and so can overheat. But when the lamp cathode coats are OK, there is no danger: Either the lamp does not ignite at all (so equal to missing lamp), or the lamp ignite, so then run up normally.
If there is only one power transistor and only one transformer, it would be the blocking oscillator type and these do instant start. However with the blocking oscillator type, the filament connection is sometimes used in the feedback path, as this type likes to fail with missing lamp. The second filament is then fed from an auxiliary winding and so when it is not conductive, it does not influence the main operation either.
This ballast style usually does not run the lamp at 100%, but it is so simple, you can quite easily build one, e.g.
this one, the transformer work quite well even on a plain ferrite rod (just recently tried that in the scope of a "simplest F4T5 from 6VDC" experiment; maybe some photos will come...)
And if you like something from the mains,
this style work quite safely (for the ballast, not as much for the lamp) with lamp in virtually any state...
The ballast will be either the 0.17A PL-S or F4..8T5 choke, or a 40W incandescent (on 230V mains; on 120V it should be 20W incandescent) But it feed the lamp by DC. But II remember operating a F40T12 on such ballast for at least five years (before the light was converted to PL-S 11W) without any visible degradation, only each year the tube polarity was swapped (during cleaning)