You would think the ACGME would make this easy to understand. They don't.
First, the chance of the program closing in the middle of your residency is slim. Programs have to be really bad, for years on end, with multiple attempts to improve, before they get closed. Or, their funding completely collapses. Both of these are unlikely.
All new programs are on "Initial Accreditation" for 2 years. They have a full site visit before they are allowed to open at all, and then after 2 years have another full site visit. The ACGME wants to make sure that they are doing what they said they would do. If there are areas of deficiency, the ACGME will let them fix them. They will change to "Full Accreditation" or "Probation" depending on their site visit, with the former much more likely -- before being put on probation, they would be given several rounds of warnings (unless they were doing something completely horrible).
"Full Accreditation" is an old term, and is being phased out.
So, for a new program, from my vantage:
Pluses:
1. The institution recently had no residents. Hence, they are used to doing everything themselves. Therefore, you're unlikely to be dumped on.
2. The faculty chosen to run the program are probably really invested.
3. Because they didn't have a program prior, they can build anything they want. They are not stuck with "this is the way it's always been done".
Minuses:
1. They are building as they go. What will your PGY-3 be like? No one knows, because no one has done it yet.
2. Will their grads pass the boards? Unknown.
3. Could the whole thing implode? Honestly, it's unlikely. The biggest worry would be the institution pulling the financial plug, not the ACGME.
4. Jobs, fellowships ... all an unknown. But probably will be fine.