I have not tried solid (closed-cell) type tyres, the general opinion of them is that they are heavy and affect bike handling. There are other opinions though.
There was a commuter on here who used something similar as a rear tyre- punctures are more likely to occur at the rear plus handling is affected less- and he was fairly happy with that solution. There may be others- it would be good if this thread flushes them out because I am curious too.
In general though few cyclists use them. Most use some puncture resistant tyres (relatively small weight penalty compared to solids), with the right tyre you should be able to reduce punctures to a very infrequent occurrence (on my commute bike and training road bike combined I am averaging less than one per year, riding in all weathers- a few thousand miles per year)