How can I get airport tax back?

At Lufthansa's Cancellation and refund page, they say (among other things)

How much time do I have to request a refund of my ticket via lufthansa.com?

You can apply for a refund online up to one day before departure. If an online refund request is no longer possible, you are welcome to telephone our Lufthansa Service Center. The standard refund deadlines apply; these will be shown in the fare conditions for the ticket you have chosen.

It therefore seems that you must request your refund through the service center.

The general terms and conditions also say

3.1.4. If you have a discounted ticket, as described in 3.1.3. above, and you are prevented from travelling due to force majeure, we will refund you the part of the airfare which, as a general rule, is non-refundable, provided that you have promptly advised us about and furnished evidence of such force majeure, and provided that the ticket has not yet been used. We are entitled to deduct an administration charge, which will be published as appropriate.

So depending on the reason for missing your flight, you might be able to get more than just the taxes back.

Further:

Article 10: Refunds

General

10.1. We will refund any unused ticket or unused portion of a ticket in accordance with the following paragraphs of this article and the relevant fare conditions:

Refund Recipient

10.1.1. The refund will be made either to the passenger named on the ticket or to the person who paid for the ticket upon presentation of satisfactory proof that the payment has been made, except as otherwise specified in the following.

10.1.2. If the ticket has been paid for by a person other than the passenger named on the ticket and if the ticket indicates that there is a refund restriction, we will offer the refund only to the person who paid for the ticket or in accordance with their instructions.

10.1.3. Except in the case of a lost ticket, we will only provide the refund once you have given us the ticket and any unused flight coupons.

10.1.4. A refund made to anyone presenting the passenger coupon and all unused flight coupons and presenting themselves as the legitimate recipient of the refund in accordance with 10.1.1. or 10.1.2. will be deemed to be a refund to the legitimate recipient.

Involuntary Refunds

10.2.

10.2.1. We will give you a refund as set out below if we cancel a flight, fail to operate a flight according to the timetable, fail to stop at your destination or stopping places, or cause you to miss a connecting flight for which you hold a reservation:

10.2.1.1. If you have not used any portion of the ticket, an amount equal to the airfare paid,

10.2.1.2. If you have already used a portion of the ticket, not less than the difference between the fare paid and the fare applicable to the segments you have already flown.

Voluntary Refunds

10.3.

10.3.1. If you request a refund for reasons other than those mentioned under paragraph 10.2.1. of this section, the amount of the refund will thus, provided the respective fare conditions stipulate as much, correspond to:

10.3.1.1. if no portion of the ticket has been used, an amount equal to the fare paid, less any reasonable service charges or cancellation fees;

10.3.1.2. if a portion of the ticket has been used, the difference between the fare paid and the applicable fare for travel between the points for which the ticket has been used, less any reasonable service charges or cancellation fees.

Refund for a lost ticket

10.4.

10.4.1. If a ticket or portion thereof is lost, a refund will be made upon proof of loss satisfactory to us and payment of the applicable fee, provided that:

10.4.1.1. the lost ticket or portion thereof has not been used for carriage or previously refunded or replaced without charging the ticket fare again (except where the carriage, refund or replacement by or to a third party resulted from our own negligence).

10.4.1.2. the person to whom the refund is made undertakes, in such form as stipulated by us, to repay to us the amount refunded in the event that the lost ticket or portion thereof is presented and redeemed by a third party for carriage or a refund, except where any fraud or use by a third party resulted from our own gross negligence.

10.4.2. If we lose the ticket or a portion thereof, the loss shall be our responsibility.

Refusal of Refunds

10.5.

10.5.1. We may refuse a refund when the respective application is made later than six months after the expiry of the validity of the ticket.

10.5.2. We reserve the right to refuse a refund on a ticket that has been presented to a country’s government officials or to a carrier as evidence of your intention to depart from that country, unless you can establish to our satisfaction that you have permission to remain in the country or that you will depart from that country with a different carrier or by another means of transport.