The way finish is being used in "Finish X soon" is as a transitive verb, i.e. one that requires an object.
To do is an infinitive. It will not be interpreted as an object.
Doing it will be interpreted as an object.
It will be interpreted as an object.
Your interpretation of "you can finish to do it soon" where to means in order to doesn't make sense semantically.
Consider "you can run to do it soon". It's a similar construction, and suffers the same problem. Soon seems off because, although it's an adverb, it doesn't imply better or different, like nicely or sweatily would. Soon can be used similarly to now (now would work because now is better than later), but it doesn't work in this context.
Replace soon with sooner and the issue seems much less. "You can run to it sooner" is a fine sentence.
"You can finish to do it sooner" is a better construction, however there is still a semantic problem.
The semantic problem might be solved with context, but it might not.
Let's assume the it in your sentence refers to something other than what is being finished, because otherwise the in order to interpretation makes even less sense. What sense is there in saying "you can finish peeling potatoes to finish peeling potatoes sooner"?
So, how about "you can finish [peeling potatoes] [in order] to [go surfing] sooner"? Well, it works, right? It's semantically similar to "if you want to go surfing sooner, finish peeling these potatoes". Or, similarly "You can finish [paying us all our money] to [get out of debt] sooner". You're offered a means of shortening the amount of time before you get something your want.
So, "you can finish to do it soon" is wrong because the intepretation is that by finishing you gain something better and soon doesn't imply improvement. "You can finish to do it sooner" is difficult to understand, so it seems ungramatical, but given the right context it makes sense.