I understand the government intends to use a number beginning 03 costing 5p per minute as a generic number for government departments in the near future.

For example, Cornwall council intends to scrap 600 numbers linking to various departments and replace them with one 03 number connecting to a central customer services which will then connect a caller internally to the relevant department.

Currently my phone package allows for free calls to 01, 02 and 03 numbers, will this affect phone packages in the future with Plusnet?

No, 03 numbers are treated the same as any other 01 and 02 geographic number. The reason that 03 numbers were issued is so companies can use them instead of 0845 and 0870 numbers which are not included in call packages and can be revenue sharing.

Ofcom regulations state that 03 numbers must be charged in the same way as 01 and 02 numbers.  Therefore, they will be charged the same way as 01 and 02 are charged in your package.

From this page on Ofcom's website (near the bottom):

"03 numbers have been available to use since early 2007. They can cost no more than a national rate call to an 01 or 02 number, and must count towards any inclusive minutes in the same way as 01 and 02 calls. Service providers including government bodies and financial organisations may choose to use these numbers instead of 08 business rate services numbers such as those beginning ‘0870’."

The 5p/min quoted probably refers to the price of a call using BT's "Unlimited Weekend Plan" where all calls are charegeable through the week.

until PN get their house in order, watch your bill on  03 call, I was charged but got it refunded

That is Essex Police, by the way