Disclaimer: I'm speaking only from my own experience and what I, personally, feel about this. Some authors may agree, others might not. I think it all depends on our mileage and what kind of relationship we have with our writing and our audience.
The short answer is: No, I wouldn't feel offended if someone commented that they are dying to know what happens next or that they're eagerly waiting for more. That's perfectly fine.
But to explain why that is I need to write a slightly longer answer. Because it's all about the phrasing and the implications of the comments that make the difference to me. The intent is, usually, not as important since, most of the time, even the comments that come across as rude and demanding are made with a compliment in mind โ the reader is trying to convey their excitement. But while I can see that on a rational level, my emotions aren't always as easy to control.
Here's an example to hopefully show what I mean (I'll even use the same emoji for fairness):
"I'm dying to know what happens next! ๐ญ"
"I'll die if you don't update soon! ๐ญ"
Both use the word "die/dying" which I think we can all agree is an exaggeration (or so I hope, at least ๐คฃ) but the phrasing changes the meaning of the comment, plus the burden of responsibility.
The first is mostly just a statement that the reader is excited to see what comes next โ which also implies that they're enjoying what they've read so far. There's a focus on the story and its content rather than just demanding a new chapter. It doesn't in any way put pressure on me as the author to write more within a certain time frame or demand that I say anything about when I might post next. The reader is just telling me that they're excited for the next chapter, whenever it happens to be posted. And, unless stated otherwise, I'll assume that the reader will patiently wait for it, no matter how long it might take.
The second, however, reads as a demand โ maybe even a threat. Because, while we all know it's an exaggeration, it's still jarring to be told that someone will die if you don't do something as soon as possible. This comment is implying that the longer I wait to post the next chapter, the worse this person will feel. I am suddenly responsible for this person's well-being and unless I post something "soon" they will suffer โ because of me. And while I can assume that the person is enjoying the fanfic due to the intensity of their comment, it doesn't feel that way to me when I read that comment. It feels like the person just wants the next chapter because they want more to read, not because they're necessarily enjoying the story or plot. It also feels like I have to explain to this person why I might not be able to post soon because they'll die if I don't.
(again, I know it's an exaggeration but it still affects me because I care about my readers' well-being and want to make them happy and, if I don't post soon, this reader will clearly not feel well)
The first puts the focus on the reader, explaining their emotions โ which is lovely โ while the second puts an expectation on me, as the author, to produce content within a certain time frame. Both may be an expression of love for the fic, but they read very differently to me.
I think a good rule when writing a comment is to simply ask yourself if you're demanding something from the author with your comment. Are you saying that you want something? Or need something? Or have to be given something? For example:
"I'm eagerly waiting for the next chapter!"
"I need the next chapter right now!"
The first is a statement, the second is a demand.
Now, admittedly, the second can be perfectly fine, too, depending on the relationship between the reader and author. I don't find it upsetting at all if that sentence is woven into a longer comment with a lighthearted, joking tone. Because then I'll assume it's more of an exaggeration and not an outright demand. But if I get a comment with only that sentence?
Then it feels like I'm a fic dispenser, not a human being.
So it can also depend on context and what else might be in the comment. It's always a good idea to say a little about the fic, how much you're enjoying it, or how it makes you feel if you want to avoid being misread as insulting or demanding. The more context I get, the easier it is for me to gauge the commenter's intent and not take offence if something happens to be a little demanding.
And I know that this might feel overwhelming because commenting is hard enough as it is, but I promise that it's mostly just a question of maybe looking over things one final time before posting and asking yourself if your comment sounds like a demand or if you're expecting something from the author by posting it.
And if you have something you want answered, asking questions is usually fine. Sometimes, that's a really good way to avoid coming off as entitled, especially if you phrase them without demands:
"Do you know when the next chapter will be posted?"
"Do you have a posting schedule?"
Both of those are totally fine to me. Granted that I might feel a little guilty when I have to answer them with "I don't know" but that's a me problem ๐ Giving an answer to a politely asked question is still a lot better than being told I'm causing someone harm by not uploading โ and making me feel like a bad person when I can't post.
So yeah. It's all about the phrasing and the context, I'd say. And where the focus is placed within the comment. If the comment demands I do something, it'll usually make me uncomfortable. But explaining what you are thinking and feeling as you're reading is fine โ it's encouraged, even!
(Well, with some limitations. Don't tell me if you hate it ๐)
And I hope I haven't scared people off from commenting. That's really not my intention. It's just that some comments are rough to read because they pile on more responsibility when I'm already tired from all the expectations I put on myself. But again, we're all different. I'm not sure if all authors are as sensitive to this as I am ๐
Thank you for the ask and take care ๐