I have two passports (neither of which is a US one). I plan to apply for a US work visa using one of my passports, and then apply for a US travel visa using the other passport. Is this allowed? I want to use two different passports because it is easier to get a US work visa on passport A, and it is easier to use passport B when travelling from one of the countries I want to visit.
In comments, you mention that the work visa is a TN visa, and that one of your citizenships is Canadian. In this case, you do not need a visa at all. To enter the US in TN status as a Canadian, you present the relevant documents at the border. Similarly, to enter in B status, you just show up at the border. There is no reason to apply for a B visa beforehand.
You can certainly get a B visa in your non-Canadian passport, but it's a bit questionable to apply for one since you don't actually intend to use it to enter the US. You will probably also have to explain why you want to use a B visa instead of your Canadian passport. If you've already established yourself in the US in TN status, furthermore, you'll have to explain why you want to enter the US as a tourist, and how long you plan to stay there without working before you leave and re-enter to resume your employment in TN status. It's mostly hard to imagine how this is less hassle than the hassle you would encounter leaving country B with your Canadian passport, or with your other passport lacking a B visa.
I say "mostly" because all of this leads me to suspect that your other passport is a country B passport, and country B prohibits dual nationality. In that case, you might want to have a look at I have two passports/nationalities. How do I use them when I travel?
Another possibility is that your other passport is visa-free in country B and your Canadian passport is not. In that case, there is no reason to hide your Canadian citizenship from country B. Just use your other passport there, but check in to your flight to the US with your Canadian passport (showing both passports whenever you need to demonstrate both your status in country B and your ability to enter the US without a visa).