Some consequences of widespread use of OCSP for HTTPS
OCSP is an attempt to solve some of the problems of certificate revocation. The simple version of how it works is that when your browser contacts a HTTPS website, it asks the issuer of the site's certificate if the certificate is both known and still valid. One important advantage of OCSP over CRLs that the CA now has an avenue to 'revoke' certificates that it doesn't know about. If the CA doesn't have a certificate in its database, it can assert 'unknown certificate' in reply to your query and the certificate doesn't work.
The straightforward implication of OCSP is that the CA knows that you're trying to talk to a particular website at a particular time. Often third parties can also know this, because OCSP queries may well be done over HTTP instead of HTTPS. OCSP stapling attempts to work around the privacy implications by having the website include a pre-signed, limited duration current attestation about their certificate from the CA, but it may not be widely supported.
(Website operators have to have software that supports OCSP stapling and specifically configure it. OCSP checking in general simply needs a field set in the certificate, which the CA generally forces on your SSL certificates if it supports OCSP.)
The less obvious implication of OCSP is that your CA can now turn off your HTTPS website any time it either wants to, is legally required to, or simply screws up something in its OCSP database. If your browser checks OCSP status and the OCSP server says 'I do not know this certificate', your browser is going to hard-fail the HTTPS connection. In fact it really has to, because this is exactly the response that it would get if the CA had been subverted into issuing an imposter certificate in some way that was off the books.
You may be saying 'a CA would never do this'. I regret to inform my readers that I've already seen this happen. The blunt fact is that keeping high volume services running is not trivial and systems suffer database glitches all the time. It's just that with OCSP someone else's service glitch can take down your website, my website, or in fact a whole lot of websites all at once.
As they say, this is not really a good way to run a railroad.
(See also Adam Langley on why revocation checking doesn't really help with security. This means that OCSP is both dangerous and significantly useless. Oh, and of course it often adds extra latency to your HTTPS connections since it needs to do extra requests to check the OCSP status.)
PS: note that OCSP stapling doesn't protect you from your CA here. It can protect you from temporary short-term glitches that fix themselves automatically (because you can just hold on to a valid OCSP response while the glitch fixes itself), but that's it. If the CA refuses to validate your certificate for long enough (either deliberately or through a long-term problem), your cached OCSP response expires and you're up the creek.