When I was young and naive, I once gave my password to an ex boyfriend and soon realized that it's a vicious and unhealthy cycle that no relationship should get itself trapped in. It's entirely unnecessary to share your passwords in a relationship and it's a sign that one or both of you are incredibly insecure and sharing passwords isn't going to fix that, it's just an outcome of that insecurity. It may make you feel comfortable for a short while but the moment you share those passwords, you're setting yourself up for disaster and future arguments. You may think it's a good thing for your relationship, being entirely honest and opening everything up about yourself to your partner but you can do that through other means.
We've all learnt by now that relationships need trust but if you can't trust your partner using their accounts, there's a problem... with you. Ask yourself why you need their password in the first place and why you feel the need to monitor something they've been doing way before they met you. You may blame it on being cheated on in the past but that doesn't justify your request. Many people have been cheated on and they rebuild themselves and their trust. If your current partner has cheated on you before, you should ask yourself why you've stayed, dealing with the constant paranoia that they'll do it again.
Furthermore, you'll get yourself stressed and paranoid, both of which aren't pleasant feelings to have and chronic stress can cause a variety of health problems. Your mood will drop, you'll become agitated easily, you may even lose sleep and they're all caused by your own actions and all make the perfect recipe for an argument.
No-one likes to lose their freedom or to feel trapped in some way, but this is how you'll make your partner feel if you request their passwords. They'll lose their freedom of privacy and if you give them the ultimatum of giving their password or you'll leave, they're going to feel trapped. The more you tighten your grasp on someone, the more they'll want to leave.
Passwords don't stay the same forever, whether that's your choice or the site itself requests that you change it for your safety. Imagine the horror of trying to log into your partner's account, only to see the words "password is invalid"! Referring back to before, you're going to think the worst and accuse them of hiding something or cheating on you. This is one situation that happened with my ex boyfriend after I chose to change my password because I wanted my 
If you start dating someone and within a short period of time, they're requesting your passwords, you should count that as a red flag - a warning for more outlandish behavior in future. Aside from most likely having serious insecurity issues, if they expect you to give up your passwords but they won't do the same, that's another red flag that they're finding ways to control you and gradually take away your freedom. That's a trap you don't want to get yourself in. If someone you've been dating for years suddenly requests your passwords but won't give theirs out, that is a huge concern. People who are guilty of something often become paranoid once they realize that, if they're capable of doing it, anyone is capable of doing the same.
If you're considering sharing passwords with your partner, please consider these points before doing so and seriously ask yourself what you're expecting to gain from it. More often than not, it will cause more bad than good for a relationship. Of course, you're free to do what you like in your relationship and you may feel that sharing passwords won't be a bad thing and you have excellent communication that you can use to set boundaries and avoid implications, that's perfectly okay.
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