I'm following a Lydiard-based approach to running -- which will result in more mileage than I was doing before and the majority of it faster than I had been running mileage so I'm a little bit concerned with the risk of injury (especially because my achilles is sore after my last tempo run). So I'm wondering what impact do you think gym workouts will have on the risk of injury during a phase of training where you are ramping up both intensity and volume? Should I cut the weights way back until I've achieved the volume I'm going for (in the end, I'm thinking 3 days a week of lifting would be good in addition to 7 running workouts) or do they actually help with injury prevention?
The training load should be modified based on recovery, as well as whatever cyclic or seasonal emphasis is important. If you are close to the limit of what you can recover from in terms of volume and intensity then something has to be dropped as you increase the mileage, or the chance of injury, illness, etc. also increases.
If you have already built a solid or adequate foundation of strength and power endurance then scale it back as you go into the period of increased mileage. Don't cut it out altogether because it is supporting in the sense of injury-prevention, as well as necessary to increase biomechanical efficiency (strong legs are always more efficient than weak legs when it comes to running). Change the emphasis of the work in the gym a bit as well because, for most, heavy strength training needs more recovery than your 7/3 schedule would allow and there's no reason to get that beat-up. So maybe visit the gym 2-3 days per week, pick up some heavy weight one of those days but keep reps low enough that you recovery quickly, then do some high-intensity, whole-body power endurance type circuit one day, and use the other for technical training, specifically to work on stuff that will help you maintain flexibility since running ruins it: OHS, Straight-leg DL, KB Swing, and other big range of motion work to tie your legs and core together and help maintain posture, i.e. Get-ups, Overhead Lunge, One-arm Overhead Lunge, etc.
Lydiard, and more recently, Maffetone, stressed a relatively long period of aerobic/distance work prior to the tempo efforts being done (and Maffetone believed anaerobic work is actually detrimental to some athletes though his point of view is 'extreme' and dependent on individual physiology and one's objective). It has been proven over and over, for years and years that one should first develop the aerobic system, and then add the tempo work, sharpening, etc. Yes, there are different ideas for aerobic development that some currently use, all of them evolving from the desire to increase training efficiency and all resulting in taking shortcuts, with the effect of restricting potential development. We (myself and some of the 'advisers' I share information with) see it over and over: aerobic deficiency results in a reduction of of the maximum speed/power one can produce long term -- not only in a an endurance event but also over years of athletic development. Your training should first maximize the power available using the aerobic system before adding any high-end work.
However, strength training improves movement economy. Some suggest it should not be included in the base-building period because of the potential negative effects on certain aerobic metabolism mechanisms (since weightlifting is essentially anaerobic in nature) but we have seen the benefits firsthand and the improvements in economy are just too great to be ignored. So we often train strength, and then explosive power in tandem with the base building or base maintenance work. It is important though to establish limits by asking, "how strong is strong enough?" The gym work is seductive and satisfying. It is tempting to do too much, which compromises development of other - and for the endurance athlete - more important systems. Typically, during an off-season build period we have endurance athletes in the gym 2-3x per week, then as sport-specific volume increases we cut back to 2x, then during the season, 1x/week. Although there are exceptions, i.e. some can sustain 2x/week in-season, we reduce gym volume across the board as the season or peak approaches for all athletes, whether runners, cyclists, or fighters.