I wouldn't put my money in to a rental kitchen but to the rest of the place, yes. Has the landlord said it's okay to paint the cabinets?

I would not paint it. It is in great shape and and looks clean. IMHO clean trumps new when showcasing property.
We have rentals with cabinets from the 80's. You know, the white ones with the light wood trim? People are lined up for these apartments because we run a clean building and fair priced building with well screened tenants.
At some point, you might want to gut and redo, but the way it looks now, I would leave it.

My impression is that the OP is the landlord, not tenant and wants to make some improvements for the young couple renting.

Much as I love white cabinets, this is not a job you should hurry through because chipping white paint over those brown cabinets would be far worse. And black would make it too cave-like, I'm afraid. It sounds as though you are in a time crunch so I'd put that into really cleaning those cabinets well. That's something your tenants will appreciate and when/if you have more time between renters, then take on the job of painting the cabinets.

I am between renters right now, getting the place ready for the new tenant.

I wouldn't recommend painting the cabinets. They will most likely get knicked and dinged (even w/the most careful tennant) and then you have a whole other set of problems. They look to be in very good condition.

The previous landlord already painted the cabinets in the brown you see, but left out the narrow door of 9 inches between refrigerator and gas stove.

I would clean it well, make sure everything is working well, and leave it as is.

I love the black and white, but since you have brown/tan countertops, I would be hesitant to go with that. Now, if you were planning to change the countertops to black or gray then I would definitely go that route.

Definitely paint.
Since you are changing the counter tops as well, I would paint in a creamy white that works with the floors, and choose a counter top in a coordinating colour.

Wash with TSP, sand lightly, prime and paint.
I haven't painted cabinets, but I understand that Cabinet Coat is popular in the US.

You sound like the kind of landlord I would like to have. :)

A really thorough cleaning would be sufficient, given that they are dirty/greast. May I ask if the room is vented, but not to the exterior, then where? Just curious.

I wouldn't paint them. If you truly feel the kitchen needs some spiffing up, what about just replacing the cabinet pulls?

Actually, I cleaned everything today and it looks so much better. The kitchen is vented through an air shaft (condo highrise). I cleaned that grate out, too, today.

Well, whoever painted them before, did a really lousy job. The cabinets are really rough, and there are brush hairs embedded. If I were ambitious, I could sand them a little and put some java gel stain over the paint and add new cabinet hardware. I tried it with the small base cabinet that was left unpainted, and it looks really nice with the gel stain.