No it won't. There are 3 Missions that you won't be able to do as part of your storyline, BUT you can do them after you finish the mission you chose.
Picking missions randomly is okay, you just have to pick according to your style. Case in point: after zero hour, you get to choose between 'unlocking' firebats or marauders. depending on your playing style, you can choose to get firebats first.
Without spoiling too much, the 3 missions don't actually affect you that much. first one doesn't really matter because both unlock the same unit, second one give you a choice over 2 units, and the 3rd is a choice over which enemies you'll face.
All in all, if you just do missions randomly, eventually, you'll end up with only one mission left. That will advance you through the story line(I think if you choose that mission before doing the others, it will warn you about it)
Hope that helps. If you don't care about spoilers, here is an in depth guide for the campaign:
In addition to all of these, there are some missions that will be easier/harder depending on which units/upgrades/mercs are available to you (e.g. Welcome to the jungle is easier if you have upgraded your missile turrets and stimpacks).