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Strictly speaking, passengers are supposed to ensure they have a sufficient balance on their card for their entire journey. It is hard to enforce, but in some specific scenario you can get a penalty fare. There are several mechanisms (including the £7 fee on new Oyster cards and the minimum balance required for entry) to mitigate abuse anyway.

The deposit / £5 fee (for cards issued before 3 Sept 2022) and £7 fee (for cards issued after 3 Sept 2022) is meant to mitigate such abuse. For older cards, a negative balance will prevent one from getting the deposit as a refund:

Strictly speaking, customers are supposed to ensure they have a sufficient balance on their card for their entire journey. It is hard to enforce, but in some specific scenario you can get a penalty fare. There are several mechanisms (including the £7 fee on new Oyster cards and the minimum balance required for entry) to mitigate abuse anyway.

The deposit (for cards issued before 23 Feb 2020), £5 fee (issued before 3 Sept 2022), and £7 fee (issued after 4 Sept 2022) are meant to mitigate such abuse. For older cards (issued before 3 Sept 2022), a negative balance will prevent one from getting the deposit / fee as a refund:

For newer cards (issued after 4 Sept 2022), the fee is non-refundable so TfL has £7 from the customer anyway.

7.1 Penalty fares apply on the National Rail network in accordance with the National Rail Conditions of Carriage. If you are within a Compulsory Ticket Area or travelling on National Rail services within the London National Rail Pay As You Go Area without: [...]

• a validated Oyster card, when you are paying as you go, showing a record of the start of your journey
• a validated Oyster card showing a record of the start of your journey and sufficient pay as you go credit, when using a combination of a Travelcard and pay as you go credit, to make a journey beyond the zonal validity of the Travelcard season ticket held on your Oyster card you may be issued with a penalty fare or you may be prosecuted.

7.1 Penalty fares apply on the National Rail network in accordance with the National Rail Conditions of Carriage. If you are within a Compulsory Ticket Area or travelling on National Rail services within the London National Rail Pay As You Go Area without: [...]

• a validated Oyster card, when you are paying as you go, showing a record of the start of your journey
• a validated Oyster card showing a record of the start of your journey and sufficient pay as you go credit, when using a combination of a Travelcard and pay as you go credit, to make a journey beyond the zonal validity of the Travelcard season ticket held on your Oyster card

you may be issued with a penalty fare or you may be prosecuted.

Short answer: You will be allowed to exit (unless it is a non-London Overground, non-Elizabeth line National Rail station). Your Oyster card balance will become negative and you will need to top up before being able to use it again.

Your Oyster card will then have a negative balance. For the card to be useful again, you need to top it up such that it has sufficient funds for your next journey. For the Tube, DLR, London Overground, Elizabeth line, and National Rail services, it is the minimum fare from your origin station (see TfL Conditions of Carriage):

• Below-18s who use Zip Oyster photocards
• People who want their Railcard discounts on off-peak PAYG journeys
• Season ticket holders
• Visitors who are facing high overseas transaction fees
• Those who are on the hunt for special edition cards

Short answer: You will be allowed to exit. Your Oyster card balance will become negative and you will need to top up before being able to use it again.

There are anecdotal experiences that this does not apply in a few non-London Overground, non-Elizabeth line National Rail stations, i.e., the automatic gate barriers will not open and you need to talk to a human at the gateline to be let out.

Your Oyster card will then have a negative balance. For the card to be useful again, you need to top it up such that it has sufficient funds for your next journey. For the Tube, DLR, London Overground, Elizabeth line, and National Rail services, it is the minimum fare from your origin station (see TfL Conditions of Carriage):

• Below-18s who use Zip Oyster photocards
• People who want their Railcard discounts on off-peak PAYG journeys
• Season ticket holders
• Visitors whose cards impose high overseas transaction fees
• Those who are on the hunt for special edition cards