How to Load a Typescript Error Log File in Visual Studio Code

I would be surprised if there's anything built-in for this. However, you can easily trick VSCode into "loading the file into the problems panel" by running a task that outputs the file contents and applies the TypeScript problem matcher to that output.

{
    "version": "2.0.0",
    "tasks": [
        {
            "label": "Load tsc.log",
            "type": "shell",
            "command": "type",
            "args": ["tsc.log"],
            "problemMatcher": ["$tsc"]
        }
    ]
}

(replace the type command with cat if you're not using Windows)

Note that the $tsc problem matcher is only applied to closed files. You could work around this by defining a custom problem matcher that reuses the problem pattern from the TS extension:

"problemMatcher": [
    {
        "pattern": "$tsc",
        "applyTo": "allDocuments"
    }
]