Cost of Increasing max pool size in SQL Server
What may be the negative effects of the increase of the max
connections pool?

If connections are indeed properly closed and disposed when not actively used (i.e. immediately returned to the pool) and you have capacity headroom on both the app server and database server, I would not expect negative effects with a larger pool size. However, increasing the max pool size to avoid errors at peak load might not be the best approach depending on the root cause. Consider the implications on both application and database server resources.
Are the queries mostly in runable status on the database server? This may indicate query/index tuning is needed or the database server could use more cores to handle peak demand within SLAs.
Are the queries on the server side mostly suspended due to async network waits? This may be a symptom the web server is slow consuming query results under load, indicating the app server might need more cores, instances, or network bandwidth. You mentioned &quot;a dedicated server&quot; so perhaps additional app servers are in order. Note that 100 active queries also requires 100 threads on both app server and database server.
Are the queries on the server side mostly suspended due to other resource waits, such as IO, locks, or latches? This can be mitigated with query/index tuning or db server configuration (e.g. more tempdb files in the case of tempdb page latch contention).