There are kind of 2 questions here:
1) Should I be worried if the company ghosts me after asking for a copy of my offer letter? Yes. You should be worried if your prospective future employer ghosts you for any reason. There are many reasons for this. One very easy reason is that, if this company establishes a practice of ghosting their employees, they might ghost you on your salary. For example, you work for a pay period (2 weeks, a month, etc), and then you don't get paid. When you ask the company for your paycheque, you don't get a response. Then what do you do? You should always be worried if your (current or future) company ghosts you.
2) Should I be woried if my offer letter is printed on recycled paper? Not necessarily. I would gauge how worried to be based on the status of the company. If it's a small company, a true "startup", then maybe they don't have resources to print a lot of documents and they use recycled paper as company policy. Maybe they're trying to be environmentally conscious and reuse recyclable materials. There's lots of explanations. However, large companies tend not to do this for professionalism reasons. If this is a large company, then I would be worried. If it's a startup, it might be an eccentricity of working there; it's certainly unprofessional of them to do but not damning.
As for what you should do from here: Here's an idea: Email the HR rep you're in contact with, and send them an email that looks roughly like this:
Hey Joe! I know we met a few days ago to sign the offer letter, and I'm still planning on coming in. Unfortunately, since I don't have a copy of the offer letter, I totally forgot what my start date was supposed to be. Do you mind reminding me which day we agreed for me to start work? Thanks.
The HR rep should respond to such a query promptly, because facilitating you coming to work on your first day is important to them. If they don't respond promptly, perhaps they're just busy. Follow up in a couple of days and see what happens. However, if the HR rep responds quickly, your level of alert should be elevated; the HR rep isn't busy, they're just choosing (for some reason) not to provide you with a copy of your offer letter. I would follow up with this:
Thanks for the response! I was just wondering, just to make sure I don't miss any of the other details of my new job, if I could have a copy of the offer letter to review on my own time before I start.
And see what happens. If they don't respond, then you should be EXTREMELY worried and should probably look for a way out of this company ASAP. Not providing you a copy of the offer letter could mean any number of things, including, for example, that they want to change the terms of your employment without telling you. For example, if you don't have a copy of the offer letter, they can modify the terms, re-collate the documents to look like you signed it, and then reprimand you or terminate your employment for a breach of a contract you never signed. This should be very worrying to you.