This turns out to be an interesting question with interesting consequences. The biggest one, which only emerged after thinking it through, is that either it can easily be worked around or all media storage is destructive even something as simple as a record player.
Ultimately, the whole thing is a lesson on why DRM can never work.
To make this answerable I'm going to assume that there's something specifically special about sending information across a network that causes it to be destroyed in storage. Otherwise any data transfer would be destructive, even displaying it on a monitor or printing it. Also it explains why you can't just write it back from memory. Why? Maybe it's weird technology. Maybe it's a totalitarian DRM scheme run wild.
This also implies that you can't copy data, or that data storage cannot hold duplicate copies. Otherwise you'd get around the problem by making a local duplicate and sending that destructively. (This is, effectively, what we do today by reading data from disk to a memory buffer before sending it to the network device).
The nature of data storage is important. If it's that you can't copy data you could "preload" a number of copies onto the same storage. This would let data servers act like vending machines that have to be reloaded from time to time giving them some utility. But if data storage is such that only one copy can exist at a time, then you physically have to reload the data every time it's send, like a single-shot rifle, making data a much more boutique thing.
For one, the Internet would never have become a consumer appliance like it is today. It would have retained its original intent: a highly robust, packet-switched phone network. Data transfer technology would have stopped at the level of modems and fax machines. The web, the cloud, streaming video, software as a service, phone apps, Facebook... none of this would have been possible or even attempted if data storage was destructive.
What would we have? We'd have anything which helps one-to-one communications, like talking on the phone. Packet-switching, for one. It is possibly the most important networking idea in he last 100 years.
When you make a call with traditional circuit switching it dedicates a fixed hunk of bandwidth for that call for the entirety of the call whether you're talking or not. This wastes a lot of bandwidth and can easily tie up the lines. It's like having a highway where only one car is allowed in each lane at a time.
Packet-switching breaks up your call into small pieces (packets) and continually sends them across the network. This means you're only using the network when you have data to send, and you're only using a small chunk of bandwidth at a time. Like cars on a highway, everyone can be streaming down the road to their own destinations.
One-to-one communication protocols would exist. Email, chat, texting, and even video chat would all exist... but only one-to-one. These are all fancy versions of talking on the phone. Data is generated on one side and transmitted to the other. The data is not stored in between. While there could be chat and mail servers like Gmail and Hangouts to gather your chats and emails while you're offline, there would be no message archiving. Once you read a message it would be deleted from Google's servers and now only exist in an archive on your computer.
No video conferencing. No group chat. No forums. No Q&A sites. No web. Anything which relies on more than one person/computer seeing the same information is not economically viable.
There would be no smart phones and high speed cellular networks. While cell phones would exist, and they could text and chat and video conference. Smart phones rely on apps which cheaply retrieve data from servers. Without cheap streaming video there would be no need to upgrade our cell networks, so we'd probably plateau at 2G speeds.
We'd have GPS and navigation, but the maps would have to be purchased and preloaded like old style car navigation devices. Geolocated services like Yelp could exist as something you purchased and preloaded onto your phone for a city you're visiting, like a digital Zagat's guide or Lonely Planet.
A cell phone with a lot of computing power and a high res screen, might exist, but it would be a hand-held gaming device that also worked as a phone. A boutique item for gamers, like the N-Gage. Cell phones would also double as a music device (if the analog hole is allowed), but you'd have to load all the music on yourself from physical media like a CD.
And music players bring us to why none of this makes sense.
That was an interesting thought experiment, but it can't work. Either all I/O is destructive, or none of it is. Why? Because what you described is basically an extreme DRM scheme, and DRM does not work.
One of the fundamental properties of computer data is that data is just data. Doesn't matter how it's stored or how it's transmitted, data is just 0's and 1's. DRM tries to make an artificial distinction between "data on your computer" and "data sent on the network" and it fundamentally does not work. The reason is The Analog Hole.
The analog hole exploits that there is fundamentally no difference between data you send to a display device, such as video to a monitor or audio to your headphones, and data you send over the network. If sending data over the network is prohibited, I can create a network device which masquerades as display device but is really a network device. If I want to copy a song, I can plug my headphone jack into a recording device. If I want to copy a video, I can plug in a "monitor" that is really a video recorder.
What if it's a totalitarian regime where all output devices must be registered and approved or they won't work? It's been tried. High-bandwidth Digital Content Protection (HDCP) is exactly that. It tries to stop protected media from playing on unapproved devices. It caused no end of frustration for legit customers, and it was cracked back in 2010. Other devices have tried similar schemes to lock down their hardware to control data transmission, phones, gaming consoles, your car... all have failed.
If your system does not close the analog hole, then the destructive nature of networking can be easily circumvented by first copying and sending it to a display device, and using that display device to transmit the copy over the network. The whole premise just amounts to working around yet another physical limitation.
If your system does close analog hole, then the consequences are extreme. DRM won't work, we know that. That means either there is something about media storage such that all reading from storage must be destructive. Something as simple as displaying a file on a monitor for the user to read will destroy that data. Music players could only play a song once, then you'd need to not just reload it, you'd need to rerecord it.
It means something as simple as a record player would play once, then be wiped clean.
Nipper, the Victrola dog, hears his master's voice once... and then never again.