I found a little more info on Google, so let me provide a partial answer to my own question. In particular, I found a post by David Wagner to sci.crypt in 2004, titled "IND-CPA for CFB mode", which in turn led me to a paper titled "Practical symmetric on-line encryption", published in FSE 2003 by Fouque, Martinet and Poupard.
In this paper, the authors prove that CFB mode (using full-block feedback) is IND-CPA secure as long as no input block to the cipher is reused (and as long as the underlying block cipher is secure), and that this holds with high probability as long as the IVs are chosen at random, and as long as the total number of $n$-bit blocks encrypted with a given key is much less than $2^{n/2}$. (In fact, with these assumptions, they prove indistinguishability under a stronger attack model which they call concurrent blockwise adaptive chosen plaintext attack.)
However, while Fouque et al. indeed assume random IVs, it does seem to me that their proof works just fine even with deterministic IVs, provided that the adversary does not get to choose the IVs and that the method of choosing the IVs does not specifically encourage collisions. (For example, using a counter as the IV should be fine, whereas choosing the next IV by encrypting the previous IV — with the same key as used for message encryption — and optionally XORing it with a predictable constant would definitely be bad.)
However, a bigger issue with the security proof by Fouque et al. is that it only provides a meaningful security margin for CFB mode with full-block feedback (and a sufficiently large cipher block size). Naïvely extending their proof to CFB-$k$ mode (i.e. the variant of CFB mode using $k$-bit truncated output with a shift register) only shows this mode to be secure when the number of blocks encrypted with the same key is much less than $2^{k/2}$. (A more careful analysis could perhaps improve this bound, but that looks like a non-trivial task.) Obviously, for e.g. $k = 1$ or even $k = 8$, this proves absolutely nothing at all!
Indeed, as David Wagner notes in his post, there are weak IVs for CFB-$k$ with small $k$, consisting of bit patterns that repeat with period $k$ or small multiples thereof. (In particular, if the IV and the plaintext consists of all zero bits, the ciphertext will also be all zeros with probability $1/2^k$.) While these weak IVs are rare, and thus unlikely to be chosen by random, some of them — including, in particular, the all-zero IV — may be likely to occur more often than by chance in a naïve counter sequence.
I'm not aware of any practical security proofs for CFB-$k$ mode for small $k$, and David Wagner also writes so in his post. However, it seems clear that, if such encryption modes are secure at all, they can only be so if the IVs are chosen at random — possibly using the NIST-recommended method of encrypting a counter — or at least in some other manner that, with high probability, avoids the weak IVs described by Wagner.
However, for CFB-$k$ mode with large $k$ (say, $k \ge 64$), the Fouque et al. proof does demonstrate security comparable to other classical block cipher modes, provided that the IVs are chosen in a manner that does not allow an attacker to easily generate collisions. Random IVs will certainly work for that, but, as far as I can tell, so should e.g. using a counter as the IV. If in doubt, though, follow the NIST recommendation and encrypt the counter just to be sure.
Addendum: I found a paper by Mark Wooding, "New proofs for old modes", IACR Cryptology ePrint Archive (2008), which gives improves security proofs for a number of classical block cipher operating modes. In particular, he writes:
"We show that full-width CFB is secure if the IV is any ‘generalized counter’, and that both full-width and truncated $t$-bit CFB are secure if the IV is an encrypted counter. We also show that, unlike CBC mode, it is safe to ‘carry over’ the final shift-register value from the previous message as the IV for the next message."
(By "generalized counter", if I read the paper correctly, Wooding simply means any fixed enumeration of the IV space, possibly known to the attacker.)