Yes, it is possible now to make tweets disappear and that too automatically after some time and all this is possible now and that too freely and what you want to understand in this case is that you are helped by the service of Spirit for Twitter.
Now this is a free service available in the web that will allow you to make certain tweets disappear after a set time. What it actually does is that it helps you make tweets get erased from your Twitter timeline for some time. To enable this, you will have to sign up for Spirit and then tweet like you do as usual.
Now what you need to understand is that if you want a specific tweet to be erased for some time, say 15 minutes, then you will have to include that specific time gap in it .Now this can be done easily and all you need to do is that you need to give a hash-tag # and then you will have to simply mention the time.
Now you have to be careful in this regard and you need to give the time in a specific format. This is something that you have to be careful about. If you want to give the time as 15 minutes, then the entry shall be #15m. In case if you want the tweet to disappear after say 2 hours, then the entry shall be #2h. And in case if you want the tweet to disappear after 3 days, then the entry shall be #7d.
Now the benefits of using this service are manifold.
First of all, you will not have to clean up the mess all by yourself. Now you never know when your timeline is going to be filled with all those unwanted tweets and so you better take the precautions.