Assign the Promise returned from yrno.getWeather(LOCATION) call to a variable.

Use Promise.all() to return results from both weather.getFiveDaySummary() and weather.getForecastForTime(new Date()) calls.

Chain .then() to the result of call to get the data at initial and subsequent .then() chained to variable identifier which returned initial Promise values.

let weatherData = yrno.getWeather(LOCATION).then(weather => {
  // note `return`, alternatively omit `return` and `{`, `}` at arrow function
  // .then(weather => Promise.all(/* parameters */))
  return Promise.all([weather.getFiveDaySummary()
                    , weather.getForecastForTime(new Date())]);
});

weatherData
// `results` is array of `Promise` values returned from `.then()`
// chained to `yrno.getWeather(LOCATION).then((weather)`
.then(results => {
  let [fiveDaySummary, forecastForTime] = results; 
  console.log('five day summary:', fiveDaySummary
             , 'current weather:', forecastForTime); 
  // note `return` statement, here
  return results
})
.catch(e => {
  // `throw` `e` here if requirement is to chain rejected `Promise`
  // else, error is handled here
  console.log('an error occurred!', e);
});

// weatherData
// .then(results => { // do stuff with `results` from first `weatherData` call })
// .catch(e => console.log(e));

An alternative to using promises directly is to use await/async.

// weather.js  
const yrno = require('yr.no-forecast')({
  version: '1.9', // this is the default if not provided,
  request: {
    // make calls to locationforecast timeout after 15 seconds
    timeout: 15000
  }
});

const LOCATION = {
  // This is Dublin, Ireland
  lat: 53.3478,
  lon: 6.2597
};

async function getWeather() {

  let weather = await yrno.getWeather(LOCATION);
  let fiveDaySummary = await weather.getFiveDaySummary();
  let forecastForTime = await weather.getForecastForTime(new Date());

  return {
    fiveDaySummary: fiveDaySummary,
    forecastForTime: forecastForTime,
  }
}

async function main() {
  let report;

  try {
    report = await getWeather();
  } catch (e) {
    console.log('an error occurred!', e);
  }

  // do something else... 
  if (report != undefined) {
    console.log(report); // fiveDaySummary and forecastForTime
  }
}

main(); // run it

you can run this (node.js 7) with:

node --harmony-async-await weather

You can use await/async on older targets by using Babel or Typescript to transpile it down for you.

Bonus (based off your comments) - I wouldn't do it this way, but just to show you it can be done:

const http = require('http');

const port = 8080;

http.createServer(
  async function (req, res) {
    let report = await getWeather(); // see above
    res.writeHead(200, { 'Content-Type': 'text/plain' });
    res.write("" + JSON.stringify(report.fiveDaySummary));
    res.end('Hello World\n');
  })
  .listen(port);

again with node --harmony-async-await weather or transpile it.