You can use ffmpeg, and you currently have a choice of four video filters that can blur:
unsharp. These filters can use the
enable option for timeline support, so you can apply the blurring effect to a certain duration if desired. You can view what filters have timeline support with
This example will apply the blur from 22-113 seconds, then again from 275-286 seconds:
ffmpeg -i input.avi -vf "boxblur=enable='between(t,22,113)',boxblur=enable='between(t,(60*4)+35,286)" -codec:a copy output.avi
You can enter seconds or have it calculate seconds for you as shown in the example.
Expression evaluation is not my best area. You may find a method to declare the filter just once.
Note that, since you only want to modify the video, the audio can be stream copied from the input to the output without re-encoding. This is faster than re-encoding and will preserve the quality. This is something most editors, such as Premiere, do not support without additional shenanigans.
If you have
ffplay you can get a preview instead of encoding, watching, re-encoding...
ffplay -i input -vf smartblur
Builds are available for Windows, Linux, and OS X users. See the FFmpeg Download page, but of course you can always compile