I'm looking at possibly buying this motherboard for use in a temporary "budget server": http://www.newegg.com/Product/Product.aspx?Item=N82E16813186144
I want to know if the memory that says it's "for use with Dual Channel DDR3 Intel Core i5 and Core i7 processors" will work with it.
You really have to trust Intel and motherboard manufacturers to keep their word on compatibility. Or you could just go to the Manufacturer's website specifically on the BlackOps motherboard and see that they support: Dual channel DDR3 1600/1333/1066/800/667 x 4 DIMMs, Max. 8GB. If i5/i7 supports these DIMMs then yes, you can use them. Otherwise, I wouldn't bet on anything and just wait until Intel and manufacturers get all the kinks worked out before buying system parts on a platform that is literally new as of today.
According to Intels own specs for the X48, it supports dual channel DDR3 RAM up to DDR3 1600. Your link to the RAM shows a DDR3 1600 dual set, so it looks correct. But note, you could also use slower (thereby cheaper) RAM, and hardly ever notice it.
So my overall suggestion would be to get pre-tested RAM for your specific motherboard model. Do this either by getting RAM that is certified by the motherboard maker, or certified for the exact motherboard model by the RAM maker, or approved by people who have the exact combination of RAM and motherboard themselves.
Anther thing is, any server should IMHO absolutely have ECC RAM. As computers are getting more and more RAM, the soft and hard error rate increases. Opinions on this are very divided, see for example Jeff Atwood's opinion or this recent study. I still think that a proper server, even a cheap low-end model from a tier 1 manufacturer, with proper ECC RAM and a Intel Xeon CPU is the way to go, even for the budget minded. Sometimes cheap DIY hardware just turns out to be way too much trouble, even for temporary use.
Yes, the 2 items linked should work fine together. The board you link to requires DDR3 1600, which is the speed of the RAM you linked to.