I believe you meant will an i5-4440 bottleneck a GTX 1080. Long answer short yes.
You don't have to get an i7 but unfortunately you will have to get new RAM, Processor and motherboard...
Why? If you are going to change the processor, you are going to have to change the motherboard since the new processors are not compatible with the old processor socket on your motherboard... They simply won't fit. You are probably using DDR3 ram, and the new mobo will only support DDR4 RAM..
You are looking at 400-500$ worth of upgrades. If you intend on gaming at 1080p, I recommend you get the GTX 1050Ti (since everything after that is super expensive atm) and make the above mentioned upgrades.
That being said, you do NOT have to make...