I have a python script that scans for new tweets containing specified #hashtags, then it posts them to my "python bot's" twitter account as new tweets.
I tested it from the python console and let it run for 5 minutes. It managed to pick up 10 tweets matching my criteria. It works flawlessly, but I'm concerned about performance issues and leaving the script running for extended amounts of time.

What are the negative effects of leaving this script running on my personal computer for a whole day or more?
Should I be running this on a digital ocean VPS instead?
Twitter offers the API for bot creation, but do they care how much a bot tweets? I don't see how this is any different from retweeting.
Twitter probably has limits on their api and will most likely block your api key if they feel that you are spamming. In fact I would bet there is a maximum number of tweets per day depending on the type of developer account.
For stability and up time concerns running on a 'personal' computer is not a good idea. You probably want to do other things on your personal comp that may interrupt your bot's service (like install programs/updates and restart). As far as load on the cpu, well if its only picking up 10 tweets per 5 min that doesn't seem like any kind of load that you need to worry about. To be sure you could run the top command and check out the cpu and memory usage.
If you have a server somewhere like at digital ocean then I would run it there just to reduce the interruption the program experiences.
I ran a similar program using twitters stream api and collected tweets using a personal computer and the interruptions were annoying and I eventually stopped collecting data....
Running python tweepy listener