In the summer of 1863 Jefferson Davis contracts yellow fever. A few weeks later Davis dies in Richmond.
Yellow fever is not the best choice for this. It comes in epidemic waves or not at all, AIUI. There were no yellow fever epidemics in Richmond during the War.
Typhoid from contaminated food (or complications of food poisoning), basically what killed Zachary Taylor, is a more likely fate.
Also, summer is from June to September. Shall we say mid-summer, i.e. late July? It's important because it's after Gettysburg and Vicksburg.
Vice President Alexander Stevens is immediately sworn in as the new President of the Confederacy.
What affects policy wise (and possibly the outcome of the war) does Davis's death have?
First off, it's Stephens, not Stevens.
As noted by others - Stephens, unlike Davis, was not a former soldier. As a West Point graduate and combat veteran (and hero), Davis reasonably considered himself suited to supervising his generals. He wasn't, which was a problem for the CSA. Stephens would have no such belief.
Stephens might name a proper commander-in-chief for the CS Army, perhaps Joe Johnston, and let him deal with the generals.
One important factor is that Davis and Johnston got into a feud in 1861 and never got out of it. Neither felt he could trust the other, and they could not discuss strategy frankly. Davis was also at odds with Beauregard.
Stephens was free of these issues.
What else? Stephens was a former Whig. Stephens was personally acquainted with Lincoln; they had been Whig Representatives together in the Congress of 1847-1848. In the crisis of 1850, when Davis opposed the Compromise, Stephens was a Unionist. He became a sort of Democrat after the Whigs collapsed, but remained opposed to the secessionist "Fire-eaters". He was a Douglas elector in the election of 1860; at the Georgia secession convention he argued and voted against secession.
His election as CSA Vice President was a sop to Georgia and to the ex-Whigs of the South.
So - what would he do as President? It seems distinctly possible that he would look at the result of 1863 and conclude that the war (which he had opposed) was lost. However, I don't think that is probable - and even if he thought so, I think most Confederate leaders would disagree. (There's a bizarre possibility - the Chief Executive is ready to give up, but the generals, governors, and Congress want to fight on. Is Stephens forced from office?) Stephens will not make any peace moves then, regardless of what he thinks privately.
Stephens was opposed to conscription and suspension of
habeas corpus, both of which were applied in the CSA earlier and far more extensively than in the Union. If he terminated those policies as President, it would substantially weaken the CSA.
OTL, by March 1864, he openly called for the CSA to try to negotiate peace. I don't think he could possibly have believed that the CSA could negotiate for its independence. That means he was proposing negotiated surrender, at least by implication.
I think that as President, he would have tried to negotiate with Lincoln soon after taking office. Unlike Davis, he would not have insisted that Lincoln treat with him as an equal. However, I can't imagine what he would offer Lincoln or ask for from him.
The only deal I can imagine being made would be for the CSA to surrender, with an amnesty for all rebels and preservation of slavery. This would mean
de facto revocation of the Emancipation Proclamation, and I don't think Lincoln would do that. He'd offered that same deal when the EP was first announced; the South turned it down.
A key question (which I can't answer) is whether Stephens would approve the transfer of Longstreet's corps from Lee's Army of Northern Virginia to Bragg's Army of Tennessee. Longstreet's forces were very important to the Confederate victory at Chickamauga.
Without them, I think the battle (which was pretty close) is a Union victory, and then Bragg is fired.
This continues the career of Rosecrans, who was fired after losing at Chickamauga, and slows down the rise of Thomas, who replaced Rosecrans, and of Grant, who was brought in to clear up the mess, and won the decisive victories at Chattanooga. Also of Sherman, who was Grant's protegé.
Getting back to Stephens: after the fall of Chattanooga, he'll be ready to propose negotiations in the Confederacy. This will be explosive, but there will be enough defeatism to let him try.
He will want to maintain slavery. Lincoln will refuse, citing the EP. ISTM that both men are under powerful but irreconcilable pressures.
Stephens wants to end the war ASAP - but if he offers to give up slavery, he would face open rebellion.
Lincoln wants to end the war ASAP - but he won't revoke the EP. However, if it's made clear that revoking the EP ends the war, there will be a lot of pressure on him not to fight a war for the slaves.
I don't know how it would work out.