Here's a simple analogy1, 1a - if you're planning on modifying the skeleton of your house, then you ought to consult with a licensed Engineer before committing to any work.
And if your renovation is altering one of the house's systems, such as the plumbing, electrical, or heating, then you can most likely rely upon the contractor performing the work and you do not need to consult with a licensed Engineer.
So what's the skeleton of a house?
It's the framework of the foundation, walls, floors, and ceilings of the building. More accurately, it is the structural components that keep your dwelling standing as well as held together. Also keep in mind that significant amounts of nearby earthwork can affect the structure of the house.
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So what counts as a modification then? It's obviously more than drilling a hole into a wall in order to hang a picture, right?
A structural modification means moving or removing a piece that was originally part of the dwelling's skeleton. For example, there might be metal posts holding up beams running the length of your basement or garage. Those posts are helping transfer forces from one area of the house into a lower area. Moving those posts would affect the skeleton of the house. Another example would be the joists between the floors of the house. Those joists help distribute the load within each floor of the house. Significant modifications to them could impact the structural integrity of the house.
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Okay, so I can't touch anything associated with the foundation or framing of the house, right?
Well, probably not. But that's Murphy's Law speaking. In other words, the thing you want to change is likely to be a critical part of your house's structural integrity. But there are non-load-bearing walls within a house. Believe it or not, those walls can be knocked out with almost zero impact to the structural integrity of the house.
Wait, what? So I can rip out the wall I want?
Slow down again. Yes, there are some walls that can safely be ripped out. There are (many) other walls where you'd be foolish to do so. How do you tell the difference?2 Hire a structural Engineer and have a proper analysis done on your dwelling and the proposed change. Reasonably speaking, the original construction of the house wouldn't have put the majority of the walls in place unless they were necessary.
Blah, blah, blah. You're just trying to perpetuate a racket and drive up Engineering services revenue.
Please keep in mind that Professional Engineers have an ethical obligation to uphold individual and societal safety over any considerations for profit. The reality is that making changes to the structural elements of your house can result in catastrophic damage to your house if incorrectly performed.
Most good contractors will avoid any potentially catastrophic project from a healthy fear of the liability associated with the project. That's a key warning sign that you should heed. That apparently expensive consult with a licensed Engineer is nothing compared to the potential expenses you would be facing to fix an errant and ill advised change to the structure of your dwelling.
Ah, c'mon. It's just a simple, stupid post that's in my way. Moving it won't affect anything. Right?
Quite the contrary. Shifting where loads are transferred can increase the amount of applied force by an exponential rate. Said another way, "simple" shifts in things can create really big problems.
Okay, wait. Wow! this is apparently very complicated stuff. What should I do?
Structural analysis is complicated, and there are a number of factors involved that have to be considered. Generally speaking, the best thing you can do is to consult with a licensed Engineer and let them know what it is you want to try and accomplish. Also let them know what you're willing to compromise upon, so they can help balance between project expenses and requirements. Their job is to help you identify the right balancing point between your dream project and the cold, harsh reality of physics. As an added bonus, they have likely already helped many other clients address similar projects. And they will likely be able to point out additional considerations that you would not have considered.
1: Please note, this answer is intentionally written in a conservative perspective. Why? Because it's the prudent approach to take when the consequences of a mistake could be catastrophic or just exceptionally expensive to repair.
1a: Also note, there are many variables at play and the range of considerations can be quite wide. There are modifications that should be answered with "Are you crazy?!" as well as modifications where the answer is "Sure, that shouldn't be an issue." This answer is meant to provide a starting point as a simple "rule of thumb."
2: Well scoped questions could potentially be asked here on Engineering or on DIY / Home Improvement. Asking about how to identify non-load-bearing walls, or asking about what building codes allow for modifications to joists ought to be on-topic for one or both of those sites.