Are visitors allowed to go deep into the forests surrounding the Amazon river?
If so, do we require any special pass from any government agency?
Visitors are allowed in the Amazon Rainforest. I have been there several times in Ecuador and Peru. I was just in Manaus last week but did not get into the rainforest.
The Amazon Rainforest is pretty much a gigantic difficult to access forest but otherwise, it is like most natural areas on the planet. Some parts are designated as National Park or Bioreserve in which case you are expected to pay, otherwise you just go in (Specifics below). In some countries such as Brasil and Ecuador, if you intend to visit the rainforest you will be asked for a proof of Yellow Fever Vaccine.
Reaching the Amazon is fairly easy but doing deep into it is more difficult. You can fly in to Iquitos or Manaus and you'll already be right next to the Amazon River. The Amazon River does not go through Ecuador but you can easily reach the Napo that feeds it with a local flight to Coca. In Ecuador at least you will also need a passport anytime you cross a port. Foreigners taking local flights also need one.
Once you arrive at your point-of-entry, most times you take a boat to get deeper. To reach Yasuni in particular, you take a boat, walk across some land and then take another boat. This must be arranged ahead of time. There are boats which take you to jungle lodges but also Amazon river cruises starting in Manaus. From there, you will take small boats to explore tributaries of the Amazon.
Yes visitors are allowed to go inside into the forests near Amazon river..but be aware because of rain all the places becomes wet and slippery..and you do not need any special permission from government agency..