I wouldn't seal a run completely with plastic, just like I wouldn't seal up a coop completely. You need to allow for ventilation.
There are advantages to a roof on a coop, if you live in an area that gets an unusually high amount of precipitation. A roof can also provide shade in the summer.
If you live in an area that gets a lot of snow, you either have to shovel, roof the run or keep the chickens confined in the coop when it gets too deep. It's not so much that you have to confine them, it's more that they won't be able to physically get out.
In the old days I had to shovel, just to get to the elevated pop hole door, to get the chickens out. I'm really at a stage in life where I just can't physically do that anymore, so now the run has a cover over the wire roof, except for the far end.
Runs should all have good drainage. Some areas get a lot more rain than others. There are times and places when protection from rain can be a good thing. Sometimes you need to get a muddy, stinky run under control temporarily and a cover can sometimes help do that. I also find that having a little of the run next to the coop under cover can get the birds outside on more of the stormy days when they would normally hang back inside the coop. Again, some areas get more frequent, heavier and stormier rains than others.
I normally have the sides to my run open. I'm thinking about adding a clear tarp to the sides next to the coop this year. I did have to shovel out the run after last year's blizzard. At this point in my life, I think I need to have a set up where that doesn't need to be done, at least not in a timely manner.
I think there are a lot of benefits to having an open run, too. Precipitation is very cleansing to the run. For awhile, we tarped the run in winter and removed it at some point in the spring. That's more work for my husband, due to the size of the run. Eventually, we decided not to do that anymore. We split the difference and left the end at the coop covered and the end away from the coop uncovered. So, only one end has plants growing. The other end has sand.
I don't think any one run is perfect for everyone. It just depends on your individual situation.