Our people who are managing databases (I can't call them DBA's because of the lack of DBA skills) are planning to take backups into the same physical disks as the database data (.mdf) files. Then all these backups are read into a tape. Although they are moving backups into the tape, I think that they should build a new dedicated physical disk for backups. This is because writing backups may cause slowness when reading databases. What do you think?
It depends on whether or not you have a performance problem, or are close to a performance problem, already. If your disks are substantially under-utilized, there's no issue and cost becomes more of a concern than performance. If there is a problem, obviously more contention will be detrimental. It's impossible to say which is the "correct" approach without knowing whether or not it will actually cause an issue, because it won't in all environments.
As Chopper3 pointed out, the availability concerns are more pressing than the performance ones, especially if you find out after a few days that, hey, the backups to tape haven't been running this week.
Assuming you have the space (this can be a very big if), it's usually not an issue.
The backups can cause a slowdown in access to the data, but that's entirely up to them. It's a matter of when they schedule the backups, and how much of a slowdown they can tolerate at those times.
Maybe it's not a prefect solution, but just feel happy they want backups!
On a side note, writing the data to the tape drive can potentially cause more of a slowdown than copying the data.
Your post indicates to me that you already know the answer. Yes you are correct this is a very bad idea on more than one level. HOWEVER if performance and recoverability of the instance and databases is not an issue to you then it wont really matter in terms of SQL Server.
One quick point to note, SQL can backup to UNC paths i.e. to another server if separate disk is not available.