Theoretical answer: yes and no.
Practical answer: not at all.
SHA-2, or specifically SHA-256, is a good hashing algorithm as far as we know. It has all the properties desired and there are no real attacks on it. It has already been battle-tested a lot in the past years. That Bitcoin uses SHA-256 makes it an even more interesting algorithm to try to crack (there is 'money' behind it now), and with all the ASIC producers popping up it becomes less work to also create ASICs that try to crack passwords hashed with this.
So with all the extra attention SHA-256 gets, it is now more likely that flaws are found rather sooner than later. However, SHA-2 is a very widely used algorithm as it is, regardless of its use in Bitcoin. If there were any great and obvious flaws, they would have been found already.
It can be compared with RSA, which now powers many of today's financial transactions and encrypted connections. That too draws a lot of attention to it, but good algorithms don't become bad just because they are in the picture.
So while all the extra ASIC production for Bitcoin mining may also provide for better password cracking tools and brings more people to look at the algorithm, this is no practical problem. Also, you should not use SHA-2 directly as a password hashing mechanism anyway. For that, see this question: http://security.stackexchange.com/q/211/10863
Edit: Read your question more thoroughly now instead of responding to the title.
Is bitcoin mining with its daily 4000+ TH/s power, funded and manned entirely by users of the hardware hoping to gain Bitcoins (and hoping they are worth some real $$) really performing a service for the NSA (or someone) and effectively "hiding in plain sight"?
No. That's just tinfoil hattery and not even worth the "thought experiment" (nice try, conspiracy theorist) since it can readily be disproved. See other answers about what we're hashing (hint: we're not finding hashes for actual passwords or other purposes).