Notes

Looking forward to:

Implies that you are already expecting to do that thing.

Usually followed by a verb, like "I'm looking forward to eating lunch."

It can also be followed by an event, like "I'm looking forward to Harumatsuri" or "I'm looking forward to the school play."

It CAN'T be followed by an object, like "I'm looking forward to a new table" <- X
Instead you would say something like "I want a new table," or if you already bought one online, or made plans to buy one, "I'm looking forward to getting a new table."

It can be followed by a noun, of the noun is something that is going to happen in the future, like "I'm looking forward to lunch" or "I'm looking forward to my flight."

If you were already talking about the thing you are looking forward to, you can say "I'm looking forward to it." For example:
"Hi, I'm Jane. I'm going to be your supervisor for the next company project."
"Pleasure to meet you Jane, I'm looking forward to it."



Longing for:

Implies that you've had a very strong desire for a long time. Without this thing, it feels like there's something missing from your life.

Must be followed by a noun, but you can use "longing to" for a verb. For example, "I'm longing for a loving mother."

Never use this with everyday objects, it sounds strange. If you say "I'm longing for a new table," it makes sense but it sounds like you are obsessed with tables. You should only use this for very special and important (or rare/expensive) things.

The expression "I'm dying to/for" is more casual and a bit less strong (which is funny because it sounds more strong lol).

It's often used for bodily needs, like "I'm dying for some food" or "I'm dying to get some exercise."
In the UK, it's quite common to hear someone say "I'm dying for a shit (クソ)" when they need the toilet, but only among friends.

You can also say "I'm dying" when you are very uncomfortable (usually from heat). For example, if you are stuck in traffic on a hot day and your air conditioner is broken, you might say "I'm dying in here."



Summary:

I'm longing for X = I really really want X
I'm looking forward to X = I'm glad X is going to happen

I'm longing for - only use for very important or rare things
I'm looking forward to - use for verbs or events, but never objects (unless the object is associated with an event, like lunch)

I'm dying for - a bit weaker than longing for, and often (but not always) used for bodily needs

I'm dying - expresses strong discomfort