Sorry for the late reply...but I hope this gets read - there is some theory about team defense that more casual fans might not appreciate.
Answer: YES, scoring would increase at first (for reason's I'll explain below). In the long term, it's hard to predict what would happen, as the dynamics of the game could shift radically to enable new strategies that exploit and defend against this rule change. So, I'll stick mostly to the near term.
WHY WOULD SCORING INCREASE: What the offsides rule does is enable the defending team's defenders to hold a line that is not far behind the defending team's midfielders (keeping the team "compact"). This makes it difficult for the attacking team to receive the ball in dangerous central positions (in behind the central midfielders) with time to turn and face the opposing team's goal. Defenders do this so there isn't much space between the midfield and defense for the attackers to play, and the defenders can step quickly to deny the turn. (This is why you see a lot of teams posses the ball with the defenders, withdrawn midfielders, and flank players...aside from counter attacks, it's very hard to play balls into forwards or attacking midfielders because they are in positions where the defense is playing very compact).
Eliminating offsides would allow the attacking team to "stretch" the field (as others have pointed out). By leaving attackers in advanced positions (such as in the corners, or near the 18 yard box), defenders would not be able to hold such a high line close to the midfield; they would have to be more cautious and play in positions to deny balls to the most advanced player. It's not so much that the deep target player would be a goal scoring machine threat, but rather, his presence would weaken defenders ability to play compact, and therefore give midfielders more time and space in the middle of the field, especially in transition. I believe this would result in a greater number of quality scoring opportunities, especially for players that are not playing in the very advanced position (i.e. midfielders and supporting attackers would score more - not the target player necessarily).
Finally, there are externalities that stretching the field create, most importantly fatigue for the midfielders. Generally, players can summon more energy when they are getting forward, but it's tougher to dig deep and track back to defend. This would result in poorer defenses and more goals.
IN CONCLUSION: While I do believe eliminating offsides would create more goals, I think the game would get uglier: very end-to-end, lots of ambitious over-the-top long balls, and would diminish the roles of creative play and technique in favor of physical gifts. While I'm not sure, I believe that over the long-term, the game would evolve to more closely resemble the flow of lacrosse: each team leaving attackers and defenders at the goals with virtually no play in the midfield except to briefly transitions between goals (and take kickoffs).