I am building a "scanner" sort of program, which reads IQ samples from an rtl-sdr stick, and keeps track of the power on certain frequencies, in certain bands. I am interested in detecting activity on frequencies, which I do by looking at changes in power. Since I'm simply calculating changes over time, can I get away with not filtering the data before calculating power, or is there some reason I should filter?
If the output from the SDR matches the band you're interested in, I see no point in filtering.
You want to filter when you have a scenario similar to this: you want to track the power in a band of $B$ Hz around a frequency $f_c$, but the SDR's output covers a band that is larger than $B$. In this case you want to filter, since otherwise a signal outside the band of interest could affect your calculations.
It is more likely than not that you would want to filter your signal to the band in which you are calculating power before making the power measurement. The case in which you would not, as described by MBaz is actually less likely in many applications: all filtering is finite, and there can be a wide variation in power level between signals of interest and signals in other frequencies. Further, these signals in other frequencies will have changing power levels over time. Note any filtering that you provide will also be finite, so the problem I describe does not get eliminated, but at least it will be understood and in your control.
So I would highly recommend filtering! It is not difficult to do, unless you know how much band selectivity (filtering) your SDR is providing, and understand that this exceeds the maximum signal strength by more than 10 dB of the total power outside of your band after the SDR provides its own channel filtering (I say 10 dB as this would result in a maximum error of 0.4 dB: $err=10log10(1+x/10)$, where x is the total undesired power in dB, an err is the resulting error in dB, but you can replace x with whatever your err criteria is to determine how much additional filtering you should provide).
Note at some point for very strong differences in power from adjacent channels, any further filtering will be of no help as non-linearity effects in your SDR will have corrupted your signal in band.