Pretty sure we can go with:
-Lyrium is NOT required in order to LEARN Templar talents, as that way is likely more academic+martial art than magical. Doesn't mean you'll be good at them when you do use them, but sure, you can learn what they are and how to do them.
-Lyrium IS required in order to TRIGGER being able to USE Templar talents
-However, Lyrium is NOT required as often as the Templars take it. It may require on the degree of Fade connection an individual has, but I don't think its a stretch to believe that some Templars may require constant dosages while others can take a small dose and do quite well for a while
-And this latter case may be Alistairs'. We can imagine him as taking lyrium at some point in some way (we know that they take lyrium at their vows but is it confirms that this is the ONLY time they can EVER start using it? Or can it just be that if a Templar hasn't taken it yet, it'll have to be at their vows?). Alistair may have a stronger magical connection than most (and for all we can think, half-elf sons of mages erm, might have to do with this), and be proof that one doesn't NEED Lyrium CONSTANTLY in order to be a Templar
So I think the only thing that's confirmed is that lyrium or a related method that ties one to a realm of the Fade is needed in order to be a Templar. One doesn't necessarily need to take it all the time, take substantial amounts, etc. Some may need this, but for many or even most others, it could be more a combination of the Chantry keeping the Templars at the top of their Anti-Magical game, and the Chantry controlling the Templars through the leash of addiction and dependance on the lyrium flowing.
Whatever is the case, lyrium allows one, mage or non-mage, to tap into the greater magics of the Fade. The mage can act as the main lyrium source and thus can direct the elements in they way they desire, while the non-mage (Templar) instead acts as a conduit of a larger song/entity's will (the Maker or whatever else), becoming more filled with purpose and the urge to protect others, and a slightly altered perception of Thedas, which can get muddled when in withdrawal and the Templar 'comes back down to earth'.
Considering that the Anti-Magic abilities that Templars use are likely among the least magically impressive abilities, it shouldn't be surprising that it doesn't take a lot of lyrium in order to make it happen, at least it wouldn't in many/most cases. Alistair keeps his abilities for a while and we can imagine the Warden either taking lyrium or having that trace of lyrium in the Joining. Beyond that, Alistair can be easily understood as someone with limited knowledge (but still a good point about lyrium dependency).
TLDR; Yes, you have to take lyrium. But its more at some point, in some amount. One may not require constant dosage in order to keep abilities for some time, and one doesn't technically need lyrium in order to learn the abilities and how to do them (though they'd still need some sort of magical/spiritual activation in order to enact those abilities). Alistair can be understood as someone making commentary on the control system of the Templars, but not denying that lyrium is important to the whole Templar process, at some point, in some amount.
In a much more magical Thedas though, one that say, doesn't have the Veil - I'm sure lyrium isn't required at all, as you'd already be magical enough yourself that you could make yourself embody the essence of 'Anti-Magic' anyway.