My 2007 GSXR 750 won't start. It turns over but it won't start up. It has set over the winter pretty much with me only starting and riding it a few times. Should I replace the battery or push start it and let the charging system recharge the battery?

If it's 54 outside and the fan is not running because the cold air is passing through the radiator and the coolant is doing it's job. Then you come up on a red light and basically all of the sudden that 54 degree wind chill has somewhat stopped (because 54 degrees outside and going 50 mile per hour creates quite a wind chill) the engine will heat up, cause the thermostat to open allowing the fluid to flow through the system. Now the fan kicks in at 220 degrees and begins to cool everything back down or you start riding and the wind cools it back down again...........I hope this helps a little. My bike heats up quickly too especially after riding for awhile and then stopping at a red light. I flush my coolant system yearly before the winter. It's generally recommended every 2 years.Ref: https://www.gixxer.com/forums/80-06-07-gsx-r600-750/294175-rapid-heating-idle.html

It sounds like your cluch pads are gone

BUY A testmeter from ebay 7 quid, run bike and check output, no more than 14 ish volts put dial to 20 anti clock wise, also feel the resin on the back of the unit, if all bubbled a sign it is shagged. test with meter pins on the battery terminals,

YOU HAVE A SHORT IN THE CIRCUT

could be the cdi or the coil u need to check ohm on the
coil and the relay they r a few things u need to do before u go and start buying parts

Check that the battery cables are secure (specifically the ground)

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Check your power commander if your bike has it. It''s the heart of gsxr.

check spark plugs,check for spark. check fuel is good. possible needle and seat gummed up in carb float bowl. disasemble and clean with brake or carb cleaner.do not change float level. re assemble ,you can use the cleaner to help to fire the motor with a shot through the carb. make sure you cleaner says flammable. good luck.

if it turns over he might want to check there isnt too much oil in the case