Regarding App Nap: the activity monitor shows NO all the time, so I am not sure, that App Nap is the problem.

Now here goes the ALL-log: 
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AllIssues & Potential Issues (4)Issues (2)Potential Issues (2)Informal (7) 
Information 
Diagnostics are performed by Remote Control Diagnostics V1.3 on a Macmini6,2 running OS X 1090. 
Searching for Hardware 
Apple® IR Receiver found on USB bus. 
Checking Drivers 
Known ISAIRDriver driver is loaded 
The IR Receiver USB device is used by Plex (PID 476) 
An application is accessing the IR Receiver USB device via an IOUSBDeviceUserClient. Depending on what it is doing, this may keep OS X and applications from receiving Apple® Remote button presses. The application is located at /Applications/Plex Home Theater.app/Contents/MacOS/Plex Home Theater. 
Candelair/Remote Buddy driver version 1.24 found 
Driver supports exclusive access 
Checking Preferences 
IR Receiver is paired to Apple® Remote ID 233 
Your Mac® will ignore button presses from all Apple® Remotes that don't have this ID. Please note that the ID of an Apple® Remote can be changed accidentally by pressing a certain button combination for a few seconds. If your Mac® doesn't react to the button presses of your remote, please unpair your remote at System Preferences > Security to exclude this as a possible error source. 
IR Receiver is enabled 
Checking running applications 
PlexHTHelper (PID 483) is using the Apple® Remote in exclusive mode 
While PlexHTHelper accesses your Apple® IR Receiver exclusively, no other application or OS X will receive Apple® Remote button press events. This can give you the impression that your Apple® Remote doesn't work. In that case, quit PlexHTHelper (located at /Users/macmini/Library/Application Support/Plex Home Theater/PlexHTHelper). 
OS X Apple® Remote support process found 
Testing exclusive access 
Can't get exclusive access 
RCD's attempt to get temporary, exclusive access failed (error: The operation couldn’t be completed. (Mach error -536870203 - (iokit/common) exclusive access and device already open) (3)). This usually means that another application is already having exclusive access to the Apple® Remote. Neither OS X nor other applications will be able to respond to your Apple® Remote at the same time. This can give you the impression that your Apple® Remote doesn't work. Quit the application that exclusively accesses the Apple® Remote to make it available to OS X and other applications again.