The simple answer is no, redundancy pay is tax free up to £30,000.
This is because it is not considered income but rather compensation for loss of position, therefore not subject to income tax.
Anything over the £30k limit should have income tax applied and deducted by your employer before they make payment to you in the usual way, but it is not subject to national insurance.
The redundancy pay does not have to all be in cash, so if you are allowed to keep your company car or laptop for example, this will be given a cash equivalent value that counts towards the £30k limit.
However, other parts of your final payoff such as unpaid holiday, bonuses due or payment in lieu of notice (‘garden leave’) will be subject to tax and national insurance in the normal way.
The amount of income tax you need to pay is calculated on an annual basis, so if you have been made redundant part way through the year it is possible that you will have paid too much tax. This is because you are taxed each month as though you will be earning that amount each month for the entire year.
The amount of tax you have to pay will also depend on the amount of income you have between the date of your redundancy and the end of the personal tax year on 5th April. It is up to you to check with HMRC what you should have been deducted and if you are due a refund or have tax to pay so do not rely on your employer to have got it right.