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Why do people have to worry about CG? Is it primarily due to fuel penalty, or take-off and landing performance, or is it mostly just to keep the aircraft from tipping over during loading and unloading, or other?
As a bonus, does tipping over of aircraft actually occur these days due to improperly managed CG?
To balance the aircraft, the sum of the lift from all lifting surfaces has to be at the same position as the weight. If the c.g. is ahead of the center of lift, the aircraft will pitch down. The pilot can counteract this by pulling on the elevator, which in effect reduces lift at the tail and thus shifts the center of lift forward. As you can imagine, there is only so much lengthwise change which can be effected with the elevator. So the first limit for the c.g. position is given by controllability.
Another important consideration is stability. You want the airplane to return to the original flight attitude if it has been disturbed, say by jolt on the stick or yoke. This can be done by creating proportionally more lift with the forward surfaces (= the wing in a conventional configuration) than with the tail surfaces. Say the jolt pitches the aircraft up. Now the aerodynamic forces should change in a way which increases lift proportionally more on the tail surfaces, so the aircraft pitches down and returns to it's original attitude. By distributing lift the way I mentioned above, the same change in angle of attack on both wing and tail will create a proportionally higher lift increase on the tail, making the aircraft (statically) stable. Thus, the second important limit for c.g. location is stability.
As for tipping over …
Effect of cg position on angle of attack stability could you please point me out where I am wrong?
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Why do people have to worry about CG?
Because the manufacturer has designed the aircraft following certain criteria, and one of those is where the CG will be during flight. If the CG is outside that interval, the elevator will not have enough autority to keep level flight.
does tipping over of aircraft actually occur these days due to improperly managed CG?
Was not improperly "managed", but it went out of bounds (in particular, the aft limit), unfortunately, soon after take-off and a 747 with 7 people on board was lost.
In flight the wing is generating upward lift, the tail plane a downward force. As you move the c.g. forward, more downward force is required from the tail plane, which means the tail plane is causing more drag, which means you will burn more fuel. Move the c.g. aft and you'll burn less fuel. However, the farther aft you move the c.g., the less stable the airplane becomes in response to gusts. As an example, 747 freighter operators typically aim for a zero fuel weight c.g. of 26% MAC (mean aerodynamic chord) as an optimal balance.
