The Constitution doesn’t say how fast the articles must go to the Senate. Some modest delay is not inconsistent with the Constitution, or how both chambers usually work.
But an indefinite delay would pose a serious problem. Impeachment as contemplated by the Constitution does not consist merely of the vote by the House, but of the process of sending the articles to the Senate for trial. Both parts are necessary to make an impeachment under the Constitution: The House must actually send the articles and send managers to the Senate to prosecute the impeachment. And the Senate must actually hold a trial.
If the House does not communicate its impeachment to the Senate, it hasn’t actually impeached the president. If the articles are not transmitted, Trump could legitimately say that he wasn’t truly impeached at all.
That’s because “impeachment” under the Constitution means the House sending its approved articles of to the Senate, with House managers standing up in the Senate and saying the president is impeached.
As for the headlines we saw after the House vote saying, “TRUMP IMPEACHED,” those are a media shorthand, not a technically correct legal statement. So far, the House has voted to impeach (future tense) Trump. He isn’t impeached (past tense) until the articles go to the Senate and the House members deliver the message.
Once the articles are sent, the Senate has a constitutional duty to hold a trial on the impeachment charges presented. Failure for the Senate to hold a trial after impeachment would deviate from the Constitution’s clear expectation.
For the House to vote “to impeach” without ever sending the articles of impeachment to the Senate for trial would also deviate from the constitutional protocol. It would mean that the president had not genuinely been impeached under the Constitution; and it would also deny the president the chance to defend himself in the Senate that the Constitution provides.