Can I update my server to 1.15 by replacing the server .jar?

I have a 1.14.4 server with the Vanilla server jar file. If I swap it for the 1.15 jar file, will there be any problems? I know the world will update okay, but will any configuration files get messed up?

It should just work, yes (of course you also need to restart the server). That's like updating your regular Minecraft game and opening an existing Singleplayer world in the new version. But it's always a good idea to make a backup before doing the update and keeping that around in case anyone finds problems later.
I play on a medium sized server that was updated to 1.15 the day after the release. The only issue that caused even this one day wait time was that a mod needed to be udpated.

More things to consider:

• New features, like bee nests, will of course not generate in old chunks, as usual.
• 1.15 does not have any major world generation changes, so no unnatural chunk borders should be visible.
• Datapacks need to be updated to format 5 to continue working. I don't think there's any mandatory changes other than simply changing the number in pack.mcmeta, but I might be wrong about this.
• The resource packs version is 5 now, but version 4 resource packs should usually still work fine. I think updating the pack properly actually requires changes, if you want to do that.
• The regular game now requires OpenGL 2.0, but I don't think this affects the server as well. Even if it did, any non-ancient CPU should have no problem with this.
• A few crafting recipes were changed, so if a datapack takes those as inspiration, you might want to adjust those recipes accordingly.
• And of course there are a lot of gameplay changes, like Nether portals being able to link to ones on top of the Nether, bees and honey being required for the advancements "Two by two" and "A balanced diet" and a lot of other smaller changes. The players should know some of those.
• There are no big new bugs that I know of*, quite the opposite, a lot of big issues were fixed.

Related post: Update existing Minecraft java world to 1.13: What to expect?

_{*Bees are not bugs, but both are insects.}