With sequential numbers, something like N04 1D34 would be nine houses away from N04 1D43.

The current system is set up so that similar numbers are not close to each other.

The second part of the numbers are semi-random so that houses near each other should not have easily-confusable Eircodes.

On top of that, sequential numbering would be a nightmare if new houses were built between existing addresses.

from the site:

The second part of the Eircode has four characters drawn from a carefully selected set of letters and numbers that identify each address. The Unique Identifier for each house or business will not be in sequence. This is to avoid the situation where a new building is created between two existing ones, and the code sequence would be broken, requiring all Eircodes in the area to be changed. In addition, as part of the process of assigning an Eircode to an address, Eircodes of physically adjacent addresses will not have similar unique identifiers; this avoids possible miscommunication or confusion, particularly in rural areas where most addresses do not have an identifying number or name.

In theory, the semi-random nature of the Eircode allows for address validation and is actually quite clever.