As far as I understand it, from reading posts by Dan Lavry (a designer of high-end hardware converters) and iZotope's Alexey Lukin, anti-aliasing should always be used, for any kind of conversion between sample rates.
The need for anti-aliasing is easier to comprehend for downsampling, since higher sample rates may contain ultrasonic frequencies that get pushed back down into lower regions if not first filtered out when going to a lower sample rate. At 44.1 khz, the highest possible frequency that can ever be represented (commonly referred to as the Nyquist frequency) is half the sample rate, or 22.05 khz. If you try to record anything higher than this, the DAC cannot distinguish between the actual signal and a lower frequency version that would fit into the same time relationship, and mathematically this works out to the difference between the ultrasonic frequency and Nyquist being 'mirrored' back down in the opposite direction. For example, if you tried to record a 32.05 khz signal at a sample rate of 44.1, it would create an alias at 12.05 khz instead. This aliased signal has no harmonic relationship to the original recording, and manifests as an unlistenable garbage tone. Aliasing was quite a significant problem in the early days of digital audio, when more primitive converter design prevented filters from being as effective at rejecting ultrasonics as they needed to be, so it is likely that most digital recordings or transfers from those days contain a hashy, distorted top end to some degree. (This may, indeed, include some of the early laserdiscs that are being preserved here).
We can see, then, that the need for anti-aliasing when downsampling is quite clear. What is less clear, however, is why it is necessary when the sample rate is increased. I don't claim to fully understand it myself, but apparently increasing the sample rate creates 'ghost images' of the original recording up in the ultrasonic range, and these too must be filtered out to maintain optimal quality. They aren't as problematic as going the other way, since the higher the target sample rate, the shallower the filter needed to keep these ghost products inaudibly low, meaning the ringing introduced by the filter will therefore be less severe. But they should still be filtered out rather than ignored—so in answer to the question posed above, I would say yes, anti-aliasing should always be enabled for SRC. It's just a matter of determining the appropriate filter slope to correspond to which rates are being used.
The ability to use shallower anti-aliasing filters is actually the true benefit of using higher sample rates. Contrary to what some would have you believe, there is really no mystical voodoo sound benefit that comes from recording ultrasonics, since not only can we not hear them, most microphones can't even pick them up, and most speakers and amplifiers will distort if forced to try to reproduce them. The sound quality within our audible spectrum of hearing is no less good at 44.1 khz than it is at 192—good quality converters will capture all available detail within that range regardless of what rate is used. Higher rates do not equate to greater time resolution at lower frequencies; in fact they are actually more prone to timing errors since the digital clock has to work so much harder to deliver each sample. Those who claim to hear sound quality benefits to higher sample rates may indeed be hearing a difference, but the cause of this is simply that there is less ringing and/or aliasing due to the filters not needing to be as steep.
From all this it is evident that the quality of anti-alias filtering is one of the most important considerations in all of digital audio, and that the best sound can only be achieved when it is properly implemented. Designing good quality filters at lower sample rates is quite a bit more difficult than at higher ones, which is why choosing the right converters (for recording, playback, and SRC purposes) is such a big deal. It affects the clarity of everything you hear throughout the entire signal chain; and of course the better your analog section, the more noticeable any problems on the digital side will become.
So to sum all that up in one sentence: always use tools with good quality anti-aliasing enabled, and you'll be far ahead of the curve when it comes to delivering high fidelity results.