If the battery is very old are there any disadvantages or danger to use notebook with power adapter only?
Short answer:

No.



Medium answer: 

There is always a risk of something failing in a fantastic way, but the chances of a deadly situation are slim to none if the user is taking good care of their hardware (no frayed cords, no liquids near electronics, etc.).



Long answer:

Simply put, batteries are arrays of cells that consist of sandwiches of two metals and a chemical between them.  The reaction draws electrons from the negatively charged side of the cell, through the electronic device's circuitry, through the positively charged metal into the chemical compound, which eventually becomes saturated as the first metal's electrons are drained.  

Recharging reverses this process, but flaws develop in the first metal's surface, causing it to resist or refuse additional inbound electron flow.  Eventually, the battery becomes unable to accept a new charge and has nothing left to give.

The battery's circuitry and charging circuitry have failsafe mechanisms to prevent shorting the battery (which can result in an explosion if there's enough current stored), shorting--and subsequently frying--the charger's power supply, and all the fun stuff in between.  With all of that safety equipment in place, it essentially means your battery's nothing but a brick that the charger refuses to touch.

So no, it is highly unlikely to explode.

All that being said, I have had old batteries swell and expand...

Check ifixit.com for what it would take to replace your laptop's battery, when the time comes.