I was trying to revive my old dirty chain for a few weeks while I waited for my replacement cassette to come in, so I partially soaked it in some chain lubricant. What's the safest way to dispose of a small amount of dirty chain lube?
Depends where you are located.
Where I live in the US the county has a hazardous waste disposal facility that takes used oil - usually from people changing car and lawnmower engine oil.
As an aside, soaking a chain in lube to get it clean will not be very effective. Clean a chain with a de-greaser or solvent (preferably a environment friendlier citrus based one), then re-lube.
Another option is to put it in a bottle and let it stand for a long time.
There's a good change the impurities will settle out down the bottom and you can skim off the good-enough cleaner at the top to reuse.
I have an old landrover that takes many litres of EP90 gear oil, and that's expensive. So when its contaminated with water, I let it stand in a big glass flagon bottle for 3 months then carefully decant. I normally get 1/2 to 2/3 back by volume, and the bottom part is disposed of as per other suggestions.