1. Learning C you will spend a fair bit of time on stuff that most
well-written C++ programs don't use (though you can generally still use it
in C++ if you want).
2. Learning C means you get into the habit of doing things a certain way and
may then be reluctant to switch to the better ways that C++ offers for some
tasks.
There is an argument on the other side:
1. Learning C forces you to become skilled in some things (notably pointer
manipulation) that are not greatly used in C++ but still need to be mastered
if you are to become really proficient as a C++ programmer. With C, you are
forced to be a little closer to the hardware, which can provide useful
knowledge.
Naturally, there are counter-arguments and counter-counter-arguments and...
Personally, I would recommend against extended study of C if your ultimate
goal is C++. However, it is a toss up whether or not you should read, say,
one C book before starting on C++. Some C++ books assume some level of prior
knowledge of C even when they claim they don't (i.e., they cover the C part
of C++ so quickly that it is rather hard to follow if you have no C
background).