The answer is, yes! A soon to be released scientific study shows that forefoot runners (those that land on the ball of their foot - see photo below) experience less impact forces in the lower extremity with a minimalist shoe.  A minimalist shoe is defined as minimal cushioning and a heel-toe drop less than or equal to 4 mm as compared with a standard run shoe.
There are several studies that reveal increased load on the lower chain whilst running/jumping on cushiony shoes and surfaces. Seems counterintuitive doesn’t it?
A study is currently being conducted specifically testing a brand that sells models that are specially highly cushioned. I won't mention the brand here but since I've seen runners wobbling around in them, I've been cranky. I’ll keep you posted.
So generally speaking based on scientific studies, cushion + more drop = more force. More force can mean greater risk for specific running injuries.
Run shoes are not the only factor on impact forces and possible running injury. I’d say they play a relatively small role in comparison. But, when you consider that running is jumping on one foot over and over again at three times your bodyweight and there are shoes that lead to increases in force sustained by the body, it stands to reason that they do matter.
So, if you are a forefoot runner should you go out and buy the least cushion and lowest drop shoes on the market and head out for a long run? If you currently wear cushion shoes, should you throw them to the curb and replace them with minimalist shoes? Ummmm....no!
For details on how to transition, check out my post: How to Safely Transition to New Run Shoes.
 Rice H, Jamison S, Davis I. Footwear Matters: Influence of Footwear and Foot Strike on Loadrates During Running. Medicine and Science in Sports and Exercise Coming Soon.
 Bishop M, Fiolkowski P, Conrad B, Brunt D, horodyski M. Athletic footwear, leg stiffness, and running kinematics. J Athl Train 2006;41 (4):387
 Zadpoor AA, Nikooyan AA. The relationship between lower-extremity stress fractures and the ground reaction force: A systematic review. Clin Biomech 2011;26(1):23-8.