Because the combat is designed around the characters finishing the campaign at level 5, you would be faced with 2 options:
- Allow the parts from level 5 onward to be easier than intended
- Tweak the encounter to make it harder
Personally I would do a bit of both. As a GM it can be easy to think that harder combat = more exciting for the players, but this is not always the case. Sometimes, having a few easy encounters will give the players a sense of accomplishment.
For example, when the players cleared cragmaw hideout, they probably found that difficult. If you throw a similar combat at them at level 4 or 5, they will just blow through it. Some players may not find that interesting but others will look back and say "wow look at how much stronger we are".
When the combats are all difficult, it can be hard for the party to see that they have made any progress.
I have done this in the past, not with the LMoP, but with other adventures. The main goal here is to compensate for the extra power you have given them. Try to analyze how hard the encounter should have been. If you are new, you could try running the combat alone (using the PC character sheets) at the intended level. Then tweak the combat, put the characters at their current level and try again.
This might seem like a lot of work, but when I was first starting as a DM I found it very useful. CR is a guideline at best. The best way to tell if an encounter is too strong to too weak (as a beginner DM) is to play it out.
In conclusion, if you do nothing with the encounters as written, the party will just find the counters easier than the designer intended. I would recommend tweaking the encounters if you plan to have party go beyond level 5.
