No!

I used to make myself complete every single episode of a drama even if I didn't like it, but not anymore.
I think it's wasted time. Why push yourself to watch something I don't enjoy when I can put those hours into something great instead? 

I love TV-series, movies, games and books, if I push myself to complete everything I don't enjoy A LOT of time would go to waste. I wouldn't have the time to discover the things I truly love if I did that... xP
At first I finished dramas that I don't like because I felt guilty when I dropped dramas. I used to skip parts I don't like. Now I only finish dramas dramas that I don't like if I'm curious about the ending or want to know if the drama redeemed himself. I usually drop a drama after 2-6 episodes if it's boring or not my cup of tea. There's no need to finish something without enjoyment. 
I always want to drop a drama if I don't like it. But it really bothers me to leave a drama unfinished that's why I try my best to understand and give the drama a chance. But if the flaws really get too much, I drop it. Like what I did to Jealousy Incarnate. Hehe. :)
someone here said to drop it in 1 episode if it's not interesting? I would have to agree/disagree with that, some dramas I think were boring/slow in the beginning but got better later. I tried to borrow some k-drama dvds from the library but they were boring (plot/script and/or uninteresting characters/actors) or too over-the-top acting so I only watched 1 episode of it. It depends on which show it is, if I want to drop or continue watching it. I couldn't stand watching MLFAS (dropped it years ago even before I signed up for this site) but I keep thinking I need to watch the rest of it just to see what's all the hype about lol
If I'm still curious enough I'd like to give the drama a chance before dropping (especially within the first two episodes). There are some really decent dramas that I might've and recently would've dropped if I didn't have a tad more patience. However, I've found that you can pretty much tell where a drama is going to go within the first 2-4 episodes, and if it's not appealing to the viewer then there's really no point in sticking with it. I have the same stategy as BeautifulSwan's, just jump to the last episode to find closure. :)
Yes, generally I like to finish drama's until it's done. Most of the reasons are all mentioned above, especially by Angelina. Mostly I'm just too curious to how a drama may end, so I end up watching them all. Although, I'd like to make an exception, because if a drama is so boring I can't stand it any longer and let's say there are more than 50 episodes. . . No matter how curious I am, I will drop it.
If there are overwhelmingly strong positive reviews and I like the actors and the  genre then I will persevere. 

Example : Nirvana In Fire. Fell asleep many times. It was all talk!  But I persevered and wala!!  Episode 19 or 20 was magic!!

Thank  god I pushed through because this drama was epic!
I force myself to finish it because I like to see the ending however bad it may be
It's like Zeigarnik effect...I keep remembering the drama if I do not finish it properly.Usually I drop dramas because it is either too melodramatic or the plot is just too poor.
These days I even skip some parts as I watch and rush to the final episode just to have peace. ^^
I have a bunch of dramas currently waiting for me to finish them.Hopefully, I will make some time for them soon.^^
If dramas get really unbearable then I stop watching them. Not really drop them (although I some some dramas on my drop list lol) but stop for a while and maybe, because there's always a possibility, I'll pick them up later when I'm in the mood to watch their genre or when I'm at lost of what to watch next. I started them for a reason so I'll eventually get the groove back and finish them and in a way, get them over with XD
I drop dramas that i like just because i dont have the time to finish them so there is no way id waste my time with dramas i dont even like
Well, I don't like to drop stuff once I start watching but I usually give it the first 2 episodes at most and
if I don't see any potential/draw/interest or something I don't continue watching and if I made it to the 3rd
episode somehow even if by the next 2 episodes I am dreading the drama I still watch it even if it takes
me forever to finish.

One of the recent dramas I had finished felt like it dragged on for eternity to get to the ending but I made it
and I finished, I don't like dropping dramas or just starting it and then just leaving it unfinished, if I start it I finish it....
I think maybe only once or twice I have dropped some dramas but I can't remember what made me drop them.

I have some dramas on hold but those are usually because of missing subtitles or a source that has all the
episodes and has them subbed, otherwise they would probably not be there unless it's like this drama called
Hooray for Love, which I put on hold for a while sometimes because it was a dauntingly long drama that kept
giving me unnecessary headaches so I would take a break from it and put it on hold and so on until I finished it.

In terms I just don't like leaving anything I started unfinished, which is why I try to make sure that I am truly
interested in something the drama has or seems to offer. Maybe I am just weird...LOL. XD
I drop really rarely! Only if I can't stand the drama or am bored to tears and have no actual care about what happens. I like to think that episodes 6 and 10 are the best dropping points--by episode 6, you can usually tell where a drama is heading and whether your wanna stay on board; and a few dramas have this episode 10 syndrome, where the whole drama just flips at that point and becomes either amazing or really boring.

I've only dropped four dramas and while I'm a little bothered by not finishing them, I also remember that I have less and less tolerance for trudging through a bad or uninteresting story. :P
I even drop dramas I like, so yeah. For me, not liking it enough is, well, enough of a reason for me to not continue. I always hesitate to add a drama to my watchlist because most of the time I won't even watch more than a few episodes.
I do finish a drama I don't like if there is a subplot or character I find interesting. This involves a lot of skipping though, sometimes whole episodes.
I drop dramas I don't like (actors that I don't know/like) at ep 1 also.
I give two three episodes or more if it has great cast. If its not getting better till 4-5 episodes then I drop it.
If I finish 10+ episodes then I force myself to finish it.
I have dropped a lot of dramas 80+