Bigint and SQL Server performance
Using bigint compared to int has, at least, these potential performance drawbacks:


the data will use more pages on disk


this can affect how long it takes to read data from disk when it's not in RAM
it will also make any maintenance operations involving those fields take longer (backups, index rebuilds, CHECKDB)

these data pages will take up more space in RAM


this means you'll either need to purchase more RAM, or incur the cost of reading from disk more often

memory grants that include these columns will be larger


this affects memory-consuming query plan operators, like sorting and hashing data
as a secondary impact, this has the effect of reducing concurrency



That being said, how much practical impact these things have for you is very dependent on your specific environment and workload.

Note: the increases in disk and RAM usage issues can be mitigated by using row compression, at the cost of increased CPU usage.  As servers often have more CPU overhead than RAM, this is generally a good tradeoff (thanks to Andy for this reminder!)