Difference between 持续 and 继续
I prefer to use 继续 in the context. Here is why : 

持续 can be construed as 保持 + 继续. E.g. 会议持续招开了两个小时 means the meeting is in the 召开 state and keep(continue with) that state for two hours. However, 生活 in this context is a generic concept and it doesn't specify what kind of life you want to persist. In other words, it doesn't specify in what way 生活 would be continuing. 

持续 would be more accurate if the sentence is put as: 不管你是快乐还是难过，"这样的生活"总是要持续下去。 It's because with 这样的生活, we clearly know how 生活 will be continuing. 

So, 不管你是快乐还是难过，生活总要继续下去 is more accurate. It means generally 生活 will 继续/continue no matter the way it is, and no matter how you feel -- happy or sad.