NFL teams generally would not welcome a proven winning coach to a division rival.
The Arizona Cardinals won't mind seeing Jeff Fisher land in St. Louis, however, because the move eliminates their defensive coordinator, Ray Horton, from consideration for the job. Horton interviewed for the position and could have been appealing relative to the other candidates beyond Fisher.
A look at other potential beneficiaries of Fisher's hiring, beyond the Rams:
Gregg Williams, New Orleans Saints defensive coordinator. Williams worked under Fisher in Tennessee. He had chances to reunite with Fisher previously, but stayed with the Saints. His contract is set to expire after this season, however. That could clear the way for Williams to leave if Fisher decides to target him for the job in St. Louis.
Lake Dawson, Titans vice president of football operations. Dawson interviewed for the general manager's job in St. Louis while the Rams were trying to land Fisher. Dawson worked with Fisher in Tennessee. A former NFL receiver, Dawson's personnel roots run back to Ted Thompson when both were together with Mike Holmgren in Seattle. Fisher's hiring improves Dawson's chances.
West Coast-based offenses. Fisher is not known for having strong opinions on specific offensive schemes. I would expect him to take into account input from Rams quarterback Sam Bradford. Bradford fared well as a rookie while running a West Coast offense. Fisher's teams in Jacksonville ran a Mike Shanahan-based variation of the West Coast scheme, similar to what the NFC West last saw when Jeremy Bates was Seattle's offensive coordinator in 2010 (Bates worked under Fisher's former coordinator in Tennessee, the late Mike Heimerdinger, when both were under Shanahan in Denver). ESPN's Mark Schlabach and others have noted that Jacksonville Jaguars offensive coordinator Dirk Koetter, the former head coach at Boise State and Arizona State, could be a consideration as coordinator for the Rams.
Fisher has had time to formulate plans in his mind. We should see signs of clear direction on the GM and coaching staff fronts before long.