I looked into the winker connections and went a bought a 3 pronged winker, because some PO had only a 2 prong hooked up and the green wire was not connected to anything.
Honda provided a two pronged flasher with the bike for US models. The extra green wire nearby was for a 3 terminal flasher where lighting requirements of other countries mandated that type.
Fuses do get warm during use. The heat from over current is what melts them when current rating is exceeded. Depending on the actual fuse type, a 15 Amp fuse should part at about 125% of rating. The Honda stock electrical load with everything on is about 10A. So, the fuse should be warm to the touch but not cause discomfort while holding your thumb on it.
My 500 has a replacement fuse holder (non Honda) on it so I don't know what the original is like. However, as Bob points out, the contacts to the fuse and the wire connections to those contacts can generate heat that conducts into the fuse, too.
Let me clarify an earlier statement. Honda convention was that black wires in the loom were switched +12V. Your replacement turn signals may follow a different convention. If the packaging instructions don't state signal color codes, it is up to you to find out what they are. You may have to dismantle the signal to learn this information.
1) determine bulb type, number.
2. verify exactly which wire color attaches to the base surround of the lamp and which drives the bulb center pin/nub.
The center pin/nub is what should connect to the Orange or Light Blue wires, whatever color code convention is used by the signal provider.
There are also bulb types that have different style of connections. I can't see your signals from here. What you have to determine is if either of the wire colors supplied attaches to a metal base of the fixture. This is important.
As someone else suggested I will try to simply unplug everything that is related to the turn signals from the system and see if the fuse still heats up with the switch on.
The Grey wire at the flasher and the Green ones in the loom, are what provides all power to the signals. If you have the proper color matched connections throughout the bike, you should get no signal illumination with the Grey disconnected. If they don't light, they probably aren't drawing power.
I have read another post about tracking down a positive power wire that is grounding out. I just want to clarify something... If I had a power wire grounding out on the chassis somewhere is that what could be causing both problems?
A "positive power wire grounding out" will cause excessive current draw to melt the fuse in very short order. To cause a slight over current that makes a fuse hot but not blow, such a condition would have to include a resistive connection only just severe as to add a small amount of extra current. I believe this would be a very rare condition, albeit possible. Such a condition is slightly more likely than all of the air molecules suddenly moving to the left side of the room. I would view reports of this nature with skepticism.
Cheers,
