Two thoughts:
1. If it's a lifter or some other valve train part, throw 2 quarts of MMO in there. It won't hurt anything.
2. It could be that you are finally getting fuel pressure to the IP. This will advance the timing a bit, so if you have an injector that's firing too soon(low pop pressure), it'll knock more than before.
I know when these IDIs run out of fuel some fuel system components can get damaged.
No, they don't. You have to worry about getting air out to get it to run again, but you won't ruin anything.
The IP uses fuel for lube, yes... but the housing(which contains the fuel used for lube) is on the return side of the pump; you will never be able to run that dry by simply running out of fuel.
IDI fuel systems are extremely hard to hurt, except with age or bad fuel(like WVO, WMO - mainly due to improper filtering).
Another thing to be aware of is that if the fuel supply side of the system is sealed(no air getting in, is primed), the IP
will run with no lift pump at all. It contains a "lift pump" of it's own, and was designed to be able to run in a gravity-feed system; it just can't prime itself if there is air in the line, or if your lift pump breaks in a manner that allows air in.
Also, a 'stuck open' injector will not just pour fuel down the cylinder. There is no pressure in the injector line until the IP fires on that cylinder. What would happen is an extremely early injection, meaning very advanced timing. At idle, it won't inject much fuel no matter what, so you shouldn't get a large cloud of smoke -- usually, smoke at idle is due to retarded timing on all cylinders, so the IP injects a lot more fuel to maintain that idle RPM, most of which isn't burning correctly.
