To start, lets do a rough estimate of how much the healing rate has been sped up.
Lets assume a week to heal the original abrasion/infection fully - 7 * 24 * 60 is 10,080 minutes. Conveniently, that means we can say the new abrasion/infection heals in (very slightly over) a minute, for a nice round 10,000x acceleration.
So what does that mean in practical terms?
We'll start with the naive approximation - assume that the magic does healing and nothing else. (And we'll conduct the experiment on a third-party, because my contract says I'm exempt from experiments on dangerous, untested magical technologies ;) )
You noted that the magic isn't providing any significant amount of energy, so the body has to do that itself. An infection like this is often slightly inflamed under normal healing, so lets assume that the body would be at least doubling or tripling the normal blood flow in order to provide the energy and material for the healing process (I.E. an extra 1-2x normal usage). That means we need somewhere between 10,000x and 20,000x the normal allocation to supply our accelerated healing.
This is the point at which warning bells should start going off.
As soon as the healing process kicks in, the tissue immediately surrounding the wound will be drained of any and all resources near-instantly. Blood vessels will dilate and blood flow will increase, but there's absolutely no way that we're going to get a 10,000x increase in blood flow. The wound and surrounding area are going to be oxygen-starved until the healing process is complete, and probably looks like a really swollen, dark-blue bruise right now.
That lack of oxygen wouldn't be a problem if the healing really was going to be over in a minute... but it isn't. The healing can't be done until the required resources arrive, and they're not arriving at anything near 10,000 time normal speed. So we've got an area of oxygen-starved tissue that we expect to stay oxygen-starved for some time. I can't find any solid references for how long it takes for cell death due to anoxia to kick in, but it seems to be somewhere between 10 minutes and a couple of hours. Unfortunately for our patient, those are being quoted for oxygen starvation due to a tourniquet, and that's going to be low oxygen; not the magically-enforced, 99.999% effective, no oxygen environment we have here, so we'll assume the low end of that range.
Fifteen minutes later, therefore, we have cells dying from lack of oxygen and energy, and that means the beginnings of gangrene... which is going to really get the magic going. At this point, we're shifting from using all of the available resources locally to using up a significant fraction of the body's total available resources. The patient's heart rate will skyrocket and he'll be gasping like he just ran a marathon, but unfortunately for him his heart and lungs are no more capable of scaling up 10,000x than his blood vessels were. We're firmly in the grip of runaway positive feedback here - any attempt at healing causes extra damage, which requires more healing, which causes extra damage...
I won't attempt to calculate how long it would take, or what the exact cause of death would be, but it's clear where this process ends. Something is going to fail, fatally, and our patient is going to wish he'd never seen that ad for clinical trial volunteers.
Wait, that's not actually healing at all!
All the above really demonstrates is that we have some Required Secondary Powers in order to get this to work correctly. One of them actually comes built-in - blood cells aren't going to be a problem, because the same processes that are accelerating cell division for healing can be easily turned to producing as many of them as we need.
A second requirement is a significant increase in the efficacy of the circulatory system in order to transport oxygen and other supplies fast enough to keep everything running smoothly. My preference here would be a combination of magical enhancement to let the blood carry far more oxygen/sugar/etc per unit volume, and an increase in blood circulation speed to help with moving enough platelets/white blood cells/etc to the injury site. (A neat little side-effect of this and the fast-repair is that the subject becomes almost entirely immune to fatigue. He'd be able to run a marathon at either 100m-sprint pace, if his lungs were enhanced, or more like 800m/1500m pace if they weren't. Same with weight-lifting, carrying heavy packs, etc - he just doesn't get tired. Ever.)
If you want actual Wolverine-style healing-in-combat, then you'll also need magical breathing- and digestion-enhancement, since that's going to need huge amounts of oxygen and stored sugars/proteins/etc for rebuilding tissue on the fly. (And the subject is going to need to eat a lot more than usual whenever there is healing going on - whether that's during the process or just afterwards to recover depends on exactly how much they're able to store.)
On the other hand, if magical stabilization and a quicker-than-normal recovery is enough, then the magic just needs to be smart about using the resources that are available - with supernatural assistance, even sealing off a major artery shouldn't be too difficult. This does mean that your super-healer can be killed far more easily, though - just stab him in the heart once a minute until he runs out of reserves.
Another thing to think about is positioning - the way an injury (naturally) heals if any cuts are stitched and bones are set is drastically different from what happens if everything is just left twisted and gaping. A pure speed-increase won't change that, nor will it decrease scarring (other than making it fade faster). On the other hand, if the magic somehow knows how everything 'should' be arranged - that is, it can work on the injury as a whole, instead of just operating at a cellular level and ignoring the big picture - then it's perfectly plausible that it would provide perfect healing. (If it doesn't account for the big picture, then a serious injury could 'heal' in a way that would cripple the patient - and the speed would mean that any bone-setting and wound-stitching would need to be done very fast)
Two potential side effects to think about are cancer and accelerated aging. Cancer is essentially cell-division-in-permanent-overdrive, which is not far off from what we have here. If the magic's control of the process is anything less than perfect, you're looking at a major cancer risk (which might be partially/fully countered by the healing, of course). Secondly, one of the major components of aging is telomere shortening, which is linked to cell replication. If you increase the average rate at which your cells replicate by a factor of ten, then you're also increasing the rate at which you age by a factor of ten.