First, are these two expressions "I feel like to .." and "I feel like ..ing .." correct?
So, What's the meaning of them?
And what's the difference between them?
"I feel like to .."
isn't something I've heard said, I don't think it's generally accepted.
"I feel like ___ing"
is correct and used to express desire.
"I feel like eating" = "I want to eat"
"I feel like swimming" = "I want to swim"
In English it's common for someone to ask "What do you feel like?" or "What do you feel like doing?" (or eating, etc). This is the general response you'd use to those questions.
This is to confirm the previous answer. We can go further and say that "feel like to..." is wrong and is always considered wrong in English. This is a huge question in English grammar and grammar books for students of English devote lengthy sections to the same question. The use of verb + to or verb +ing depends always on exactly which verb you are using the first. In the case of "feel like" +ing is required.
TL;DR "I feel like to" is rarely used, and is slang, and definitely wouldn't be used as a substitute for "I feel like ...ing." Trying to compare the two is like comparing apples to oranges (they are both sentences, but the similarities end there).
"Feel like to" is a slightly tricky phrase to use, the most common form being along the lines of "What/Why/How would/does it feel like to ...," a type of question that would typically have an answer in the form of a first-person story that describes what your senses would feel, or what thoughts would be running through your head, as events unfolded.
I have heard "I feel like to ..." orally spoken, and very infrequently written, such as this use from the Angry Video Game Nerd:
I feel like to beat the boss you have to either get really lucky or have no life in order to perfectly memorize all possibilities...
"I feel like to" does exist in some slang dialects, with the intent of expressing the speaker's thoughts or beliefs, but those who prefer proper English would avoid this particular construct. I would not intentionally advocate trying to use "I feel like to ..." in regular conversation, because it will almost certainly be grammatically incorrect. Even the quoted sentence would be more appropriate written as "I believe, in order to beat the boss, you have to..."
I believe that this question may have revolved around the OP's understanding of the trio of trick words "to", "too", and "two," possibly orally, and mistakenly thought "to" was being used instead of one of the other words. Alternatively, it may have been written, in which case, the writer almost certainly used the wrong word. This trio of words is among the top 10 "most confusing sets of words in English," and even many native speakers mix them up regularly. The other two words could normally appear with "I feel like ...", such as:
I feel like two cups of sugar is enough.
I feel like too many fingers spoils the pie.
You can't say, for example "I feel like to eat", unless it was an expression of a thought, as just mentioned, where you might say "I feel like to eat cookies, they have to be served with milk." In most cases, a more desirable statement would express the thought directly, such as "I believe that in order to eat cookies, they must be served with milk."
Since "I feel like to" has such limited usage, you would typically use "I feel like ...ing" in normal dialogue, which expresses a desire to perform the verb in the sentence in a "passive voice". "I feel like dancing" means that the speaker has a desire to dance, but it does not imply intent. If the sentence were instead formed with an "active voice," the sentence would be "I'm going to dance," or something similar.
This passive form is perfectly acceptable in response to a question, such as: "What do you feel like doing tonight?" "I feel like seeing a movie." When not in response to a question, you should avoid "I feel like" because it is a weak, passive statement.
By way of example, you wouldn't normally walk into a room, look at someone, and say, "I feel like making a sandwich". Instead, you would most likely say "I am going to make a sandwich." Usually, such a declaration of intent would include an offer to make additional sandwiches for anyone else that might happen to want one (at least, in American society).
I feel like eating
is perfectly correct. It means I want to eat. Most verbs can be substituted for eating if you put the -ing ending on them:
I feel like shopping - (I want to shop)
I feel like going - (I want to go)
I feel like fishing - (I want to fish)
I feel like [verb]ing - (I want to [verb])
Your other construction (I feel like to eat) doesn't work but can be altered to be:
I feel like I want to eat - (I think I want to eat)
I feel like I am going to be sick - (I think I am going to be sick)
I feel like it will rain today - (I think it will rain today)
I feel like [clause] - (I think [clause])
In this case feel like means think.
I feel like an ice cream - (I want an ice cream)
I feel like lunch - (I want lunch)
I feel like a beer - (I want a beer)
I feel like [food or drink] - (I want [food or drink])
When talking about food or drink I feel like is another way of saying I want.
I feel like a bore - (I feel boring)
I feel like a failure - (I feel unsuccessful)
I feel like a king - (I feel wonderful)
These are actual feelings. Not at feelings can be expressed this way, e.g. I feel happy would be I feel like a happy person as a happy by itself is ungrammatical.
The phrase "I feel like X" needs X to be, or act like, a noun:
To fit a verb into this slot, as we often want to do, it needs the "-ing" form as a kind of adaptor, making it act like a noun. Technically this is called a gerund. So although "walking" isn't a tangible real-world noun (you can't have a pile of walking), grammatically to a native English speaker it feels like "a thing" and it fits quite happily into the "feel like" phrase:
I'm not a native English speaker, nonetheless how I understand the difference between
"I feel like eating"
"I want to eat"
Is that for the former the action of eating is desired. Whereas the latter is not necessary the action of eating, but may also refer to the result of eating - full. -ing is an ungoing process, whereas withing -ing it may refer to a single point in time, not (necessarily) a reference to a point where you are at the table - eating.