In the long run, the game seems to severely favor starting whichever class gives you the best proficiencies and no incentive whatsoever to take Wizard at level 1. If you are planning to start from level 1, however, there might be a couple of early-game drawbacks:
Some classes get some very nice (read: expensive) equipment as part of their base package, typically because such equipment is necessary for their core functionality. Obvious examples that come to mind are chainmail and the Wizard's spellbook. If you start your primary class, you'll be fully kitted out to perform that function. If you start your secondary class, then when you hit level 2, you'll be needing to do some shopping.
Yes, 50 gp (cost of a spellbook) isn't a ton of money, even at level 2. But depending on where the campaign starts, you may not be able to access large-ticket items, even assuming you were able to round up the necessary cash in that time. Until you can correct that, you'll be short a major aspect of your primary class's functionality. Plus you'll need to account for how your spellbook starts with 300 gp worth of spells (6 spells @ 50 gp each) already copied into it.
This really hits the Wizard hardest, but any caster might suffer from the lack of a focus (a caravan or mining town isn't likely to stock that kind of thing), while front-line fighters would probably miss the heavy armor and weapons they would otherwise have.
This isn't really a hard detriment, but depending on your DM could make for a rough start. Particularly, based on your earlier question, if they balk at giving you a 50 gp leg up as part of your character's background.
The PHB mentions level 5 as a major change in the tier of play, citing extra attacks and level 3 spells. I would argue level 3 is very similar, in that it doesn't really feel like you're playing the class you chose until you hit level 3. Primary casters in particular double the number of spell slots they get, including level 2 slots. As a level 2 caster/level 1 anything else, you'll be lacking a significant amount of oomph relative to single-classed characters. That's 18 medium difficulty encounters (or 12 hard ones) where you'll only get 4 spells per long rest.
However, once you hit level 4 or so, these things will have largely evened out and your Rogue start will leave you two skills and 2 hp ahead, picking locks and flashing rapiers with the best of them in your fancy studded leather armor.