Fan noise/overheating is a common complaint of the i7 NUC. I think you would notice the i7 runs significantly hotter and louder than your i3. This is because the i7 is a hotter chip than i3 (28w vs 15w TDP) therefore the fan will need to work a lot harder.
Here is a relevant video demonstrating the cooling capabilities of the i7 NUC in action:
You should be able to just move your hard drive from i3 to i7 (I would 1st uninstall some of the drivers, like video), however you may need to re-active Windows 10 (and it may not work, if you used the free upgrade to Windows 10, unless you have a Windows 10 product key).
I have had my NUC i7 for five months, using it everyday and I can hear the fan when the load on the processor goes above 50%. I like the size of NUC but I would not recommend it. Even surfing some of the web pages with tons of stuff on it, brings NUC i7 to its knees.
I would return the i3 and wait for the 6th Gen NUC Skylake (at least two months away), or wait for Intel 5x5 motherboard.
Welcome to "the sea of disappointment" as one Community member has dubbed our little NUC owner's club.
For what it's worth, I returned my NUC i7 but have not yet successfully found a tiny-form-factor replacement that fits my needs. Maybe the technology I am looking for just doesn't exist yet; suggestions? For the time being I am simply using a cheap laptop, which while noticeably slower-performing than the NUC, does manage to run cool, quiet, and stable.
The (6th Gen) Skylake NUCs with Core i3 are already shipping. The Core i5 version is supposed to ship by the end of the year. Note, however, that it will likely take two weeks longer for units to filter down through the distribution channels and get onto store shelves.
If you want a box this small, with this level of computing power, you pay the piper keeping it cool. I can't hear my NUCs except when I am running something intensive (of course, I am 58 and, while I supposedly have normal hearing for someone my age, it's certainly not going to be as good as you youngsters have). If yours is really loud, there might be something wrong with the cooling unit...
Thank you for contacting the Intel® Communities.
Regarding transferring of your drive from one unit to another, that may lead to software like issues, as OS key related issues mentioned in previous answers.
My recommendation when changing to one computer to another will be performing a good and clean OS installation.
I really like using partitions in my builds, it helps with data-save and avoiding 3TB backups that last an important amount of time (no matter the system).
Regarding the cooling, lets remember the size of the unit, also where it is located, make sure the vents of the fan are not being covered by anything that may affectt he airflow.
Since this is a small form factor solution, the cooling may not be as in a full size computer, lets keep that in mind.
There are different "profiles" at BIOS level for the cooling solution, you could test them and which would help you better.