Forrest’s death, while detrimental to the morale of his units, will not really alter the course of the war aside from giving the Union a slightly easier win.
From June 1863 to October 1863, Forrest served the Army of Tennessee as a cavalry division and later cavalry corps commander. Forrest did not excel in these capacities. Forrest might have been a tactical genius and a badass; but he was also insubordinate, and essentially useless at performing reconnaissance and screening for an army.
He fell for Rosecrans’ diversions during the Tullahoma Campaign and failed to properly screen the Confederate advance in the run-up to Chickamauga. To elaborate on the latter, Forrest erred in not securing the crossing over the Chickamauga, not watching the right flank of the infantry with videttes, not screening the Reed's Bridge road and allowing it be set on fire by Union troops, violating the chain of command when he borrowed a division of the reserve corps without authorization, and not preventing or seriously obstructing Gordon Granger from reinforcing George Thomas on Horseshoe Ridge during the battle itself. In the pursuit to Chattanooga, Forrest delivered inaccurate intelligence to Bragg that the Union army was evacuating Chattanooga, when it was actually digging in. Forrest’s death is not really a negative for the CSA but possibly a positive if his replacement is more competent in this role.
Forrest’s death will be more felt at Mississippi and Tennessee, but they were now relatively strategically irrelevant. Forrest’s raiding made life for Union garrisons dangerous and railroad men frustrating, but Forrest never truly achieved any major strategic impact aside from diverting A.J. Smith’s wing of the XVI Corps to Mississippi. Even his greatest victory, Brice Crossroads, could be judged as an extremely costly success for the Union as Sturgis’ job was to distract Forrest from raiding Sherman’s lifeline during the Atlanta Campaign. With a less competent commander, the only difference is that Mississippi feels the hard hand of war more and Tennessee less so, and that A.J. Smith would be sent to besiege Mobile months earlier than OTL.