The lyrium is what allows them to exercise magic negating abilities that enables them to combat mages more effectively. So long as they keep taking the correct dosage, they can maintain their abilities and stay sane. If they stop taking it they start suffering withdrawal symptoms and unable to function properly mentally. They may not be drooling idiots but they certainly can't think straight. All Templars are aware of this and so would always ensure they have an adequate supply with them when embarking on a journey. Obviously when at home base they can simply get it from stores, or may be it is just routinely issued to them at meal times.
Previous games have conveniently ignored the necessity for lyrium when the PC trains as a Templar. May be the powers we get are lower level ones that can be achieved simply by the correct training and discipline of the mind, without the need for lyrium. Certainly Meredith and Evangeline seemed to be able to stop a mage's spell in its tracks by touching them and channelling their power into them, which I don't think is the same as anything the PC can do and so would appear to be higher level ability. Perhaps this time round the Templar spec will clarify this for us.
Cullen is a long term Templars, as is Cassandra, so they will be addicted to lyrium. However, they have probably not yet been taking it for long enough for the degenerative effects to set in. They could be at risk, though, so may be there will be some sort of plot linked to this, possibly with us discovering a way to cleanse them of lyrium addiction, much in the same way that Fiona was cleansed of the taint, and then they are able to operate their powers simply by virtue of the amount of lyrium already in their system, or something similar.