I, personally, do not ban LINQ usage in my company - but rather encourage its use whenever appropriate. LINQ is very expressive, and I find that it produces code that is often much more maintainable, and higher quality, than "traditional" methods of development.
In addition, I often find that many developers tend to write more performant code using LINQ than "traditional" looping methods. The streaming nature of LINQ can dramatically cut down on the runtimes if used correctly.
Personally, I find that any company that "bans" the usage of a specific language or framework feature outright has higher level problems. Every technology exists for a reason - and every feature has its place. Whether it's appropriate in a specific scenario is another issue - but banning it completely is a sign of poor management of developers, in my opinion.
If not, what applications does your company develop and why is performance not so crucial?
Performance is absolutely critical to me. My company develops scientific software, and many of our routines have runtimes in the hours (or even days), so every ounce of performance we can squeeze out is very useful.
That being said, performance doesn't come from micro-optimizing in most cases - it comes from designing a better architecture and "large scale" algorithms. The key for this to be truly successful is having a good design, and keeping the code as simple as possible. Simplicity helps in profiling tremendously - which in turn can pin point the real performance issues in the application.
Occasionally this will be due to a LINQ statement, but very rarely. More often this is due to a poor design decision. I find that LINQ actually reduces the frequency of poor design decisions. It's much easier to refactor a LINQ statement into a more performant tight loop when necessary than it is to try to profile an application that's "chattier" in terms of code than necessary.
In addition, LINQ has some huge performance opportunities, even on a small scale. For example, it's much simpler to parallelize a LINQ query via PLINQ than to try to parallelize many other constructs, for example. Granted, it's not magic, and care still needs to be taken, but LINQ tends to parallelize with fewer issues than loops. If performance is the goal, simple, clean code should be the target, and LINQ helps achieve that more quickly.