Bigger question is: would you drive with a blindfold on? That’s how it is if you develop a system without referential constraints. Keep in mind, that business requirements change, application design changes, respective logical assumptions in the code changes, logic itself can be refactored, and so on. In general, constraints in databases are put in place under contemporary logical assumptions, seemingly correct for particular set of logical assertions and assumptions.
Through the lifecycle of an application, referential and data checks constraints police data collection via the application, especially when new requirements drive logical application changes.
To the subject of this listing - a foreign key does not by itself "improve performance", nor does it "degrade performance" significantly from a standpoint of real-time transaction processing system. However, there is an aggregated cost for constraint checking in HIGH volume "batch" system. So, here is the difference, real-time vs. batch transaction process; batch processing - where aggreated cost, incured by constraint checks, of a sequentially processed batch poses a performance hit.
In a well designed system, data consistency checks would be done "before" processing a batch through (nevertheless, there is a cost associated here also); therefore, foreign key constraint checks are not required during load time. In fact all constraints, including foreign key, should be temporarily disabled till the batch is processed.
QUERY PERFORMANCE - if tables are joined on foreign keys, be cognizant of the fact that foreign key columns are NOT INDEXED (though the respective primary key is indexed by definition). By indexing a foreign key, for that matter, by indexing any key, and joining tables on indexed helps with better performances, not by joining on non-indexed key with foreign key constraint on it.
Changing subjects, if a database is just supporting website display/rendering content/etc and recording clicks, then a database with full constraints on all tables is over kill for such purposes. Think about it. Most websites don’t even use a database for such. For similar requirements, where data is just being recorded and not referenced per say, use an in-memory database, which does not have constraints. This doesn’t mean that there is no data model, yes logical model, but no physical data model.