The Railpass (Interrail Eurail is your ticket) you can buy the mobile pass at anytime even just minutes before the start (You still need time to regist your Railpass with your ID-Card/Passport)

Reservations (that sometimes must be paid on top of the Railpass) depends from Country to Country :)
Paperreservations via Interrail atleast 9days in advance
E-Reservations via Interrail atleast 3hours in advance
Reservatns at stations or online at the operator website often till last minute :)

Some Trains just have limited seats for Railpassuser like Eurostar to from London, TGV from France to Spain or Thalys between Paris - Brussels/Amsterdam/Cologne and are often sold out weeks in advance in high season like summer or christmas :/

The prices for ONLY the SEAT (or bed/sleeper) REServation not.

But if you mean (you are not very clear about this) tickets without pass-often yes, this depends per country.

Normal tickets (not Eurail/Interrailpass) usually increase closer to the travelday :)

Only in France, a domestic seat reservation can become more expensive (from 10 to 20 EUR), depending on how busy the train is.

Other reservations don't become more expensive but can sell out, especially from/to France and Italy.