The error you're getting is what you get when you try to push to a frozen array:

const temp = Object.freeze([]);
temp.push(42);

You've shown that you're passing the array to DataGrid as rows. Apparently, DataGrid freezes the array, presumably because it needs to know that the contents of it don't change.

If you want to change those contents, you'll need to store temp in state and re-render after adding to it; see *** comments (I've also renamed temp to dataGridRows):

function Room() {
    const [textInput, setTextInput] = useState('');
    // *** Store it in state
    const [dataGridRows, setDataGridRows] = useState([]);

    const handleTextInputChange = (event) => {
        setTextInput(event.target.value);
    };

    const handleSubmit = () => {
        console.log('here goes');
        if (textInput !== '') {
            const refer = ref(database, 'rooms/');
            push(refer, textInput).then(() => {
                setTextInput('');
                toast.success('Added Successfully!');
            });
        }
    };

    useEffect(() => {
        const refer = ref(database, 'rooms/');
        onValue(refer, (snap) => {
            snap.forEach((child) => {
                console.log(child.val() + child.key);
                // *** Add to it in state; this will cause a re-render
                // so DataGrid picks up the change
                setDataGridRows(dataGridRows => [...dataGridRows, { id: child.key, firstName: child.val() }];
            });
        });
    }, []);

    return (
        <div>
            <Grid item xs={12}>
                <SubCard title="Room List">
                    <div style={{ height: 400, width: '100%' }}>
                        <DataGrid
                            rows={dataGridRows}
                            columns={columns}
                            pageSize={5}
                            rowsPerPageOptions={[5]}
                            components={{
                                Toolbar: CustomToolbar
                            }}
                        />
                    </div>
                </SubCard>
            </Grid>
        </div>
    )
}

Thanks for the answer TJ Crowder, it helped me to understand that the real problem is not the object that you are going to pass the value, but the original object, because it is passed by reference.

So, when you have the error

object is not extensible at array

The solution is to copy the values of the original object to the target, not just assign them with the equals (=) operator.

for example

targetObject.nodes.map( (newNode: any) => {

    //here I create a new object and enrich it with an array and a boolean
    let newFullNode = Object.assign({}, newNode, {nodes: []  , hasChildren:true}); 
    
    targetObject.nodes.push(newFullNode);
});

I update the targetObject later on, and when I was updating it, only then I was getting the error, which was misleading me.