Scenario A: You have a system that solves problem X. Scenario B: You have a system that solves problem X and has to ensure that it is always synchronized with the redundant backup.
It is pretty clear that the system in Scenario B has to perform strictly more work than the system in Scenario A, therefore the redundancy has a performance impact.
However, note that this is not necessarily always the case. It depends on how the redundancy is designed, what it is designed to protect from, and so on.
An example that is probably familiar to many of us: RAID 1 vs. RAID 5. RAID 1 achieves redundancy by mirroring. This means everything you write to disk is also written to all other disks as well. So, the write performance is the minimum of the individual write performances of all drives. But reading can be done in parallel (assuming that you are not actually in the disaster case at present), so the read performance is actually the sum of the individual read performances of all drives.
Oh, but wait: it actually isn't. Not quite, at least: because if you rely on that fact for your system to work, then it is no longer redundant! As soon as you lose one drive, your read performance will go down. So, your system has to be designed to be able to work with that performance. If it is designed to work with being able to spread the reads across all drives, then it is no longer redundant. And if you have 7 mirrored drives, but your system requires the read performance of 6 drives, then your RAID is actually only 1-redundant, not 6-redundant as you would a 7-drive RAID 1 normally expect to be.
So, while it turns out that RAID 1 does not necessarily have a negative performance impact, and in fact, can even boost performance, actually relying on this performance boost will negate the redundancy.
RAID 5, OTOH, spreads out both writes and reads across all drives and uses an EDAC code (error detection and correction) to be able to reconstruct data. This means you have the slight performance overhead of computing (or reconstructing) this EDAC code, but OTOH, you get a performance boost also for writes. But again, as soon as you are in the failure case, your performance drops, and if you rely on the performance, then you don't have redundancy.
So, RAID 1 is an example of a redundancy scheme that has practically no negative overhead in the worst case, and can speed up one of the two operations (read but not write) in the best case. RAID 5 is an example of a redundancy scheme that has a slight overhead in the worst case (reconstructing the lost date from the EDAC code), but speeds up both read and write in the best case.