What are the main problems that may arise if all countries close (or almost close) their borders for 1-2 years or more?
It makes sense that once the spread of the virus is under control in an area that barriers are then established to isolate that area. In practice this is simplest at a national level, because we have national governments. Nations with a federal structure, US for example, can perhaps do so at the state level. In certain cases, and dependent on local geography and politics, these areas can then later be widened, thinking about a single Australia/New Zealand zone for example.
But closing of borders doesn't have to mean no entry, it can - and likely will - just mean quarantining. Any people coming in from an area where the virus is still rampant will need to be quarantined for a period of time (a couple of weeks is I think what China is using) to ensure they are virus-free when they enter. For goods/produce coming in from abroad, there will likely be strict decontamination procedures.
Border closure seems prudent to me. It may cause delays in physical cross-border business, it may severely hamper the transport of perishable goods (especially foodstuffs and certain medicines), and it will likely mean huge reductions in international travel of people, but in the absence of both a vaccine and - crucially - the capability and willingness to administer such a vaccine globally, then border restrictions are an obvious and sound defensive strategy.
So long as some part of the world has the virus, everyone is vulnerable. We all need to work together. Whilst there is always huge opposition to any idea of supranational governance (hello the EU), those concerns now need to be put aside. It has been demonstrated many times that we will not work together out of any sense of moral duty or compassion to less fortunate nations, but we now find ourselves in the strange situation of being compelled to work together out of basic self-interest.