Emulator

Interestingly, if you stay on that black screen and disconnect the Wii remote, an in-game prompt will appear to tell you that communication with the Wii remote was lost. If you reconnect the Wii remote, the game removes the prompt but continues to stay on the black screen.

Can you verify that you have a good dump of the game? I hate to ask this first and foremost, but it's one of the most common reasons for a game to hang like this.

Sorry, is there a specific way to check? I got my dump using USB Loader GX from my Wii using my own physical copy. Should I just do it again and see if that makes a difference?

you can verify the disc integrity in dolphin in the game properties menu.

I am assuming you are referring to the MD5 checksum. I clicked "Compute" and I got this value: c131209291a5537efc45ac5e50533cd0. I also tried having Dolphin extract all the files in the ISO to a folder, which was successful.

I just checked the Dolphin Wiki page for this game and under "Problems" it mentions that area warping does not work 95% of the time. I searched and did not see this issue reported by anyone else, but it seems I am not the only one who ran into this problem.

The MD5 doesn't match GameTDB (http://www.gametdb.com/Wii/SFVEXJ), but I don't completely trust them, plus hashes are less useful for Wii games than they are for GameCube games - Wii games are often scrubbed and trimmed (unless you're ripping them yourself), which cuts out mostly useless data and other disc partitions (Wii update partitions, SSBB's Masterpieces (!), etc.) but usually leaves the game partition itself untouched (aside from lossless compression).

Wii discs contain integrity verification data in each partition, so while the MD5 might be near-useless, you can still check the game partition's integrity: right-click the game > Properties > Filesystem > right-click a partition and click "Check Partition Integrity". If it says it's bad, go redump it with CleanRip instead of USB Loader GX. http://wiibrew.org/wiki/CleanRip If it passes... well, redump anyway, just to be thorough.

Alright, here's what I did:

First, I checked the partition integrity on my existing dump and it passed.

Next, I redumped using CleanRip. I had Dolphin compute the MD5 Checksum and got this value: 3d1c9d4f0a88e3b1fb00c24159473996. It matches the one on GameTDB.

I then checked the partition integrity on both partitions and they both passed.

I booted up the game and tried to fast travel again and the same issue still occurs.

Since I have verified that there is nothing wrong with my game dump, I decided to upload a YouTube video that shows the issue: http://youtu.be/HXeAD74Mx2A

I apologize for the audio, but for some reason it started stuttering when I hit the record key in FRAPS. The video shows me using the (upside-down) mini map to fast travel. Once I select a location, the game fades to black permanently. A few seconds after the fade to black, I use the Alt+F5 shortcut to disconnect the Wii remote so that you could see the in-game prompt appear.

I played for about 2 hours without problems with the rev 5586 (x64) with default settings. ;)
The rev is downloadable from the official dolphin-emu.org website. ;)

Unclear why this is still open. It was reported as fixed /w r5586, and I can confirm it works in 4.0-8325.

Other comment from wiki editor for posterity:
"Pretty sure this was reported by people who tried to warp to locations they hadn't explored, when this is done the game will accept the 'destination' but just load the same location again. I explored 3 of the starting locations and was able to work consistently between them all without any issue, if I tried to warp to areas without flags I was just warped to the fishing shop."