Should I hold my breath when I do resistance training exercises?

Breathing is good.  Always promote good breathing. :)  Rule of thumb during resistance training is to breathe in on the negative and breathe out on the positive.  I know a few people who do not like the loud grunters in the weight room, however, the louder you are, then you know you are breathing out at the right time.  So it's not altogether bad.  Again breathing is always good.  Holding your breath is always bad. If you don't like being the loud person in the weight room you can use whatever breath sounds you would like.  I just breathe regular, while I have taught some to use sounds from the alphabet such as the "s" sound.  It is up to you.

If you have a tendency to hold your breath while working out or lifting weights it is possible you could be placing increased pressure on vessels and other structures that are not designed to hold that kind of pressure.  In addition, no one wants you to pass out.  That also is bad.  Holding your breath can cause you to get dizzy (not safe in the weight/work out room).  It also doesn't allow your muscles to be properly oxygenated.

An example of good breathing would be if you are doing a bicep curl: breathe out on the way up and breathe in on the way down.

You should never hold your breath when working out.  Now this is a common thing that I see a lot in gyms.  While most people don't do this on purpose it is something that should be avoided.  When you think about it when you are working out your body needs oxygen to workout for you muscles.  So if you deprive the muscle of oxygen while working out you will find exercise will be much harder to do and also can cause fainting.  So just use controlled breathes when doing resistance training to avoid these  problems. 

No, it is not advisable to hold your breath when you do your resistance exercises.  Holding your breath intentionally during resistance exercise is called the valsalva manuever, named after the doctor who first described it.  Whether intentional or not, holding the breath during resistance exercise can lead to large spikes in blood pressure, can cause hemorrhaging of the eyes, and can lead to immediate fainting.